# Hawkes autoregressive processes: a new model for multiscale and heterogeneous processes

Théo Leblanc

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Théo Leblanc. Hawkes autoregressive processes: a new model for multiscale and heterogeneous processes. 2025. hal-05361674

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# Hawkes autoregressive processes: a new model for multiscale and heterogeneous processes

Théo Leblanc¹,ᵃ

¹CEREMADE, Université Paris Dauphine - PSL, Paris, France, a leblanc@ceremade.dauphine.fr

**Abstract.** Both Hawkes processes and autoregressive processes depend on linear functionals of their past while modelling different types of data. As different datasets obtained through the recording of the same phenomena may be heterogeneous and occur at different timescales, it is important to study multiscale and heterogenous processes, such as those obtained by combining Hawkes and autoregressive processes. In this paper, we present probabilistic results for this new Hawkes autoregressive (HAR) model, including the existence of a stationary version, a cluster representation, exponential moments and asymptotic behaviour. We also derive statistical results for estimating interactions, extending the well-known LASSO estimation method to Hawkes Autoregressive (HAR) processes.

**Résumé.** Les processus de Hawkes et les processus autorégressifs dépendent tous deux de fonctionnelles linéaires de leur passé tout en modélisant des données de natures différentes. Différents jeux de données issus de l'observation d'un même phénomène peuvent être hétérogènes et peuvent se produire à différentes échelles temporelles, il est donc important détudier des processus multi-échelles et hétérogènes comme celui obtenu en combinant des processus de Hawkes et autorégressifs. Pour ce nouveau modèle Hawkes Autorégressif (HAR) nous établissons des résultats probabilistes, notamment l'existence d'une version stationnaire, une représentation en cluster, l'existence de moments exponentiels ainsi que le comportement asymptotique. Nous obtenons également des résultats statistiques pour l'estimation des interactions, en étendant la méthode d'estimation LASSO aux processus Hawkes Autorégressifs (HAR).

MSC2020 subject classifications: Primary 60G55, 62M10; secondary 60G10, 62H12

Keywords: Hawkes processes, Autoregressive processes, Cluster representation, Stability, LASSO estimation

# 1. Introduction

## 1.1. Motivations

Hawkes processes, introduced by Hawkes in 1971 in [29], are self-interacting point processes modelling the continuous arrival of points over time. In the multivariate case, points are emitted by several nodes. The occurrence of each node may alter the future probability of a point being emitted by any node. Hawkes processes can be self-exciting, self-inhibiting, or both simultaneously. This flexibility enables Hawkes processes to model a wide range of phenomena, such as earthquake occurrences [46], financial markets [24], epidemiology [43] and the activity of the neural network of the brain [9, 26, 38, 49], to cite but a few. Concerning applications in neuroscience, Hawkes processes model a network of neurons where each node is a neuron and the points of a given node correspond to the spiking times of the associated neuron. With this modelling point of view, the dependency of the process on the past occurrences models, in particular, synaptic integration. Hawkes processes have been the focus of very active research in the past decades, see for example [3, 12, 19]. Autoregressive (AR) processes, as opposed to Hawkes processes, are discrete time processes with continuous values. However, Autoregressive processes depend on their past as Hawkes processes, and this is why Hawkes processes are often referred to as autoregressive point processes. Indeed, the value at time $t \in \mathbb{Z}$ of an AR process depends on a function of the previous values at time $s < t$ and on an additive noise term, [10, 53]. Depending on the function that codes the past dependency, AR processes may be self-exciting, self-inhibiting, or both simultaneously, as for Hawkes processes
[34]. Moreover, as for Hawkes processes, AR processes have a multivariate extension, often denoted VAR or MVAR, for (multivariate) vector autoregressive processes, where several time series depend on each other and on their past [42]. We call these discrete random sequences chains in the present article. The simplicity and versatility of AR processes make them popular and widely used in many fields, such as economics and finance, see [7, 25]. More recently, AR processes have been used in many other fields. One such field is neuroscience, where EEG (electroencephalogram) data and brain rhythms can be modelled using AR processes [39, 59].

Hawkes processes model continuous arrival times of events whereas AutoRegressive processes are able to model discrete time series of real values, which makes Hawkes processes and AR processes highly complementary since they have similar past dependency while being of different nature. Since experiments may produce simultaneously different type of data it is natural to seek for heterogenous models, i.e. combining processes of different nature such as a continuous time process and a discrete time process. Another important observation is that data may exist on different timescales, for example if two simultaneous phenomena are recorded with different frequencies of measurements. Thus, it is natural to look for a model also able to handle multiscale data. The main goal of the present article is to introduce and study a new model combining a Hawkes process and an autoregressive process with a multiscale architecture.

Hawkes processes have already been combined with other processes. Indeed, in [21] and [1], the authors combine a diffusion and a Hawkes process to create an interesting heterogeneous model where points of the Hawkes process are introduced in a stochastic diffusion. However, the diffusion does not interact with the Hawkes process; therefore, only one cross interaction is taken into account. Hawkes diffusion models have many applications in finance [30].

Multiscale processes naturally arise in the context of wavelets and lead to AR processes defined on trees [5, 18] or autoregressive modelling for each component of the wavelet transform [37]. However, AR processes on trees are often non causal, as they introduce interactions between ascending or descending scales regardless of causality [5, 18]. It is also possible to introduce a multiscale architecture into Hawkes processes or in AR processes by parameterising interactions with varying memory parameters, see [2] for Hawkes processes and [4] for AR processes. These models are, however, homogeneous since they do not combine Hawkes and AR processes. In the present article, we instead adopt the approach used in [54]: to introduce a multiscale nature to AR processes we consider a multivariate AR process $X = (X^1, \dots, X^P)$ where each time series $X^k = (X_t^k)_{t \in \tau_k Z}$ has its own temporal embedding $\tau_k Z$ where $\tau_k$ is the period.

A multiscale and heterogeneous model was introduced in [54] to describe the coupled interactions between the activity of individual neurons and brain rhythms, leading to a complete description of brain activity and functional connectivity. The model mixes discrete time Hawkes processes and multiscale AR processes. This new model is interesting in itself as well as for the perspectives it brings to neuroscience. In the present article, we aim to build upon this by relaxing the discrete nature of the Hawkes process. To this end, we introduce a model consisting of a continuous time Hawkes process and a multiscale autoregressive process in reciprocal interactions, named HAR process for *Hawkes Autoregressive process*. In [54] the authors mostly develop the modelling side, the discrete time nature making the mathematical work quite straightforward. Having a continuous time Hawkes process instead of a discrete one reveals all the theoretical difficulties which were not addressed in [54] such as points accumulations in finite time. More importantly we obtain sharper conditions for the existence of the process in continuous time. The next section explains how to naturally combine a Hawkes process and an autoregressive process to obtain a Hawkes Autoregressive (HAR) process.

## 1.2. Heuristic for HAR processes

We provide an heuristic to combine Hawkes and Autoregressive processes in order to obtain HAR processes. For this heuristic we only focus on the one dimensional case.

**Hawkes processes:** A Hawkes process $S$ is defined through its stochastic predictable intensity. The intensity at time $t$ can be viewed as the infinitesimal probability of having a point between $t$ and $t + dt$ while knowing the past before $t$,

$$ \lambda_t dt = \mathbb{E}[dS_t | \mathcal{F}_{t-}]. $$

For Hawkes processes, the intensity depends on the past as follows:

$$ \lambda_t = \phi \left( \int_{-\infty}^{t-} h(t-s) dS_s \right), $$

where $\phi$ and $h$ are two functions. Unlike $h$, we assume $\phi$ to be non negative and non decreasing. The function $h$ models the self interaction of the point process $S$ on itself. The function $\phi$ allows for the introduction of nonlinear dependency,
[12, 29]. Thus one can write

$$ (1.1) \qquad \mathbb{E}[dS_t | \mathcal{F}_{t-}] = \phi \left( \int_{-\infty}^{t-} h(t-s) dS_s \right) dt. $$

**Autoregressive processes:** For AR processes, the current value of the observation can be calculated through the past ones, up to some noise. AR processes can be defined by the following equation

$$ (1.2) \qquad W_k = \sum_{r \ge 1} a_r W_{k-r} + \varepsilon_k, $$

where $a_r$ are fixed coefficients for positive integers $r$ and $\varepsilon_k$ is a random noise independent of the $W_{k'}$ with $k' < k$ [10, 53]. The function $a: r \in \mathbb{N}^* \mapsto a_r := a(r)$, where $\mathbb{N}^*$ stands for the positive integers, models the past dependency. One can generalise AR processes by allowing a more complex past dependency leading to

$$ W_k = \Psi(W_{k-1}, W_{k-2}, \dots) + \varepsilon_k, $$

with $\Psi$ a function. In order to obtain the analogue of (1.1) we stick to the following level of generality,

$$ (1.3) \qquad W_k = \psi \left( \sum_{k' < k} a(k - k') \kappa(W_{k'}) \right) + \varepsilon_k, $$

where $\psi$, $a$ and $\kappa$ are given functions. Taking $\psi = \kappa = \text{id}$ leads to the classical AR model described in (1.2). The function $\psi$ allows the introduction of some nonlinearity and is the analogue of the function $\phi$ in (1.1). The function $\kappa$ allows for more flexibility and even more nonlinearity, for example, one can use $\kappa(x) = |x|$ or $\kappa(x) = x_+$.

By denoting $\mathcal{D}$ the counting measure on the set of the integers, (1.3) becomes

$$ (1.4) \qquad W_k = \psi \left( \int_{-\infty}^{k-} a(k-k') \kappa(W_{k'}) d\mathcal{D}_{k'} \right) + \varepsilon_k. $$

In particular, we obtain from (1.4) the following equality:

$$ (1.5) \qquad \mathbb{E}[W_k | \mathcal{F}_{k-1}] = \psi \left( \int_{-\infty}^{k-} a(k-k') \kappa(W_{k'}) d\mathcal{D}_{k'} \right). $$

General Hawkes and AR processes have very similar structures, see (1.1) and (1.5). So it is easy to combine them. For this purpose we add cross interaction terms in the intensity equation and in the AR equation, leading to the following system of equations.

$$ (1.6) \qquad \left\{ \begin{array}{l} \lambda_t = \phi \left( \int_{-\infty}^{t-} h(t-s) dS_s + \int_{-\infty}^{t-} h_{W \to S}(t-k) \kappa_S(W_k) d\mathcal{D}_k \right) \\ W_k = \psi \left( \int_{-\infty}^{k-} a(k-k') \kappa(W_{k'}) d\mathcal{D}_{k'} + \int_{-\infty}^{k-} h_{S \to W}(k-s) dS_s \right) + \varepsilon_k, \end{array} \right. $$

where $h_{W \to S}$, $\kappa_S$, $h_{S \to W}$ are some functions. Equations (1.6) provides the heuristic definition of HAR processes.

### 1.3. Probabilistic contributions and related works

In the present article, we focus on deriving the following probabilistic results on HAR processes: existence of stationary HAR processes; cluster representation; exponential moments; and, finally, rate of convergence of HAR processes to the stationary distribution, leading to coupling à la Berbee and concentration inequalities. These probabilistic results are crucial, in particular, to prove theoretical guarantees for estimation procedures.

**Stationary HAR processes.** Brémaud and Massoulié [12] have shown existence and uniqueness of stationary Hawkes processes if the spectral radius of the matrix gathering the $L^1$ norms of the interaction functions is smaller than 1. An
AR(∞) process with autoregressive coefficients $a_k$ for $k \ge 1$ is stationary and causal if for example $|a_1| + |a_2| + \cdots < 1$, condition, which also extends to a spectral radius condition in the multivariate case, see [42]. We extend both these results by stating existence and uniqueness of stationary HAR processes under the condition that the matrix gathering the $L^1$ norms of the interaction functions (including both homogeneous and cross interactions) has a spectral radius smaller than 1, see Theorem 3.9. To prove this result, we use a Picard iteration, which is a classical method in the Hawkes framework [12].

**Cluster representation.** The cluster representation for linear Hawkes processes was introduced by Hawkes and Oakes in [31]. This representation is a powerful tool for performing in-depth calculations, leading to results such as mixing properties [8] and exponential moments [40]. In Theorem 4.5 we derive a cluster representation for linear HAR. This extends the cluster representation of linear Hawkes processes. We obtain this cluster representation by solving the autoregressive equation to decouple the equations. HAR clusters are generated by bivariate functions $h(s, t)$, depending on both the past occurrence time, $s$, and the present time, $t$, whereas for classical Hawkes processes we only consider $h(s, t) = h(t-s)$, see (1.1). Let us mention that in [52], the authors studied locally-stationary Hawkes processes: a collection of point processes indexed by $T > 0$ with cluster functions given by $h(s, t/T)$ and base rates depending on $t/T$, with a focus on the dependence on $T$ of the moments and Laplace transform. Limit theorems have been proved for similar processes where $h(s, t/T) = g(t/T)\phi(t-s)$ in [20]. For HAR processes, we are far from the framework of [20, 52] since cluster representation for HAR processes leads to 1-periodic bivariate functions satisfying $h(s, t) = h(s+1, t+1)$. The clusters we obtain, are a particular case of clusters, for which sharp bounds on their exponential moments are derived in [41].

**Exponential moments.** In [40], exponential moments for Hawkes processes were proved using a cluster representation under minimal assumptions, namely sub-critical clusters, which is equivalent to requiring the spectral radius of the off-spring matrix to be smaller than 1. Previously, exponential moments for Hawkes processes were derived with strong assumptions on the decay of the interaction functions, see [28, 50], or with strong conditions on the offspring matrix such as requiring its spectral norm to be smaller than 1. In Theorem 5.3, in order to prove exponential moments for HAR processes under the condition of sub-critical clusters, we borrow and adapt the ideas from the proof in [40] to our case involving the autoregressive part. Since the clusters are generated by bivariate functions depending on both $s$ and $t$, we also use the results recently developed in [41] about these type of clusters.

**Rate of convergence and stability.** Given a dynamic, for an initial condition $C_0$ and $N^{C_0}$ the process following the dynamic with initial condition $C_0$, the dynamic is said to be stable at $C_0$ if the process $N^{C_0}$ shifted by $t$ units of time converges in distribution as $t \to \infty$ to the stationary process $N$ following the dynamic, see [12]. Brémaud and Massoulié [12] proved stability for multivariate nonlinear Hawkes processes under spectral radius condition for existence of the stationary process. This result is only a convergence in distribution and more precise statements arise from the study of renewal times and regenerative properties for Hawkes processes, see [17] and [27], allowing to decompose Hawkes processes in independent random cycles and revealing a recurrent Markov process. By using a convolution equation, Clinet and Yoshida [16] even proved convergence rates; in the case of exponential Hawkes processes, the $L^1$ convergence rate is also exponential. In Theorem 6.3, we prove similar results for HAR processes. More precisely, for exponentially decaying interaction functions, we prove that two HAR processes, constructed on the same randomness with two different initial conditions, converge almost surely toward each others as time goes to infinity. We also prove exponential convergence rates in $L^p$.

**Coupling and concentration possibilities.** Our last probabilistic contribution is about a coupling and applications to concentration for HAR processes. Concentration results for Hawkes processes can be achieved using various methods, such as renewal times [17], mixing properties [8] or Kalikow decomposition [47, 48]. For HAR processes, however, we turn to a coupling method, à la Berbee, inspired by [6]. This idea has been widely used to obtain concentration results for stationary time series see for example [23, 51, 58]. For Hawkes processes, the idea is to introduce a coupling consisting of independent pieces, and such that the coupling is sufficiently close to the initial process, see [50]. To construct the coupling we use non stationary HAR processes started at different times. For example, a HAR process $X^{t_0}$ started at $t_0$ is independent of the past before $t_0$ and, due to the asymptotic limit discussed above, is close to the stationary process if it has evolved for a sufficiently long time. Thus the pieces constituting the process look like $(X^{t_0})_{|_{[t_0+\ell,t_0+2\ell)}}$ for $\ell$ large enough, see Definition 6.5. This coupling is used, in particular, in the statistical part, Section 7, to prove concentration of the Gram matrix.

## 1.4. Statistical contributions and related works

In the present article, in addition to the probabilistic study of HAR processes, we propose a statistical method based on LASSO to estimate the interactions of HAR processes. LASSO was introduced by [56] and consists in minimizing a penalized contrast by an $L^1$ penalty. We refer to [13] for general review.
**Related works about estimation for Hawkes processes.** Numerous techniques have been developed to estimate the interaction functions of Hawkes processes. One example is Bayesian inference, where concentration rates are derived for the posterior distribution leading to non-parametric estimation of the parameters and consistency in the graph adjacency of the interactions, see [22] in the linear case and in nonlinear case, [55]. Another possibility is to use the maximum likelihood (MLE), as in [9], where the authors first estimate the adjacency graph of a nonlinear (ReLu link function) Hawkes process with MLE and thresholding, and then re-estimate non-zero interaction functions with MLE. About the adjacency graph of interactions of a linear Hawkes process, in [15] the authors recover a superset of it, by estimating the covariances. Estimation based on least squares and the LASSO has been introduced for a large class of point processes including Hawkes processes in [28]. The authors restrict the parameter space to linear combinations of a fixed dictionary of functions with bounded support and derive an Oracle inequality for the LASSO estimator of the interaction functions of a linear Hawkes process. Further work has been carried out in the linear case such as [2] where the dictionary has a tensor structure and in [36] where the base rates depend on covariates and global parameters. In [36] the authors also provide a debiasing method. In the nonlinear case, especially for Hawkes processes with ReLu link functions, LASSO estimator have been the subject of [14] with strong assumptions on the intensity and [38] with applications in the recovery of functional connectivity in neural networks in the brain.

**Related works about estimation for Autoregressive processes.** Estimation of parameters for autoregressive processes (AR) and more generally MA, ARMA and ARIMA processes is discussed in the review [35]. Various techniques exist, such as MLE, as discussed in, for example, [39], with applications to EEG data, or ordinary least squares (OLS). When the OLS is penalised with an $L^1$ penalty, a LASSO estimator is obtained. In [45], the authors provide convergence rates for the LASSO estimator of the coefficients of an AR($p$) process with $p = O(\log(n))$, where $n$ is the number of observations. The rates depend on the penalty weights, but are approximately in $\sqrt{p/n}$. In the multivariate case, for vector autoregressive processes, the authors of [32] use the LASSO to select and estimate non-zero coefficients in a faster way than with other model selection procedures. They also prove a convergence rate of order $1/\sqrt{n}$ for the LASSO estimator.

**LASSO estimation for HAR processes.** To estimate the interaction functions of linear HAR processes we use the LASSO procedure to select non-zero parameters and achieve sparse estimation, even when the dimension is large. This approach was also adopted in [54] for discrete time HAR processes. We combine both classical least squares contrast for linear Hawkes processes and for autoregressive processes and an $L^1$ penalty with adaptative weights. Adaptative weights, see [33, 60], take into account the heterogeneity of the coefficients, thereby improving the performance of the LASSO. Here, as in [28, 36, 54] we allow the weights of the $L^1$ penalty to depend on the data in order to estimate the variability of each coordinate of the estimator. Following the steps of [28, 54], we also introduce a dictionary of functions with bounded support and prove an oracle inequality in Theorem 7.6. This oracle inequality is stated under two conditions: (i) a control of the noise on each coordinate, to calibrate the LASSO, and (ii) a restricted eigenvalue property for the Gram matrix, which is a classical condition to obtain oracle inequalities [57].

**Calibration of the LASSO.** Following the method described in [28, 54], we derive data-dependent Bernstein-type inequalities for HAR processes to control the heterogeneous level of noise. In Theorem 7.9 we deduce weights that are measurable with respect to the data, and which fulfill with high probability condition (i) for the oracle inequality to hold.

**Restricted eigenvalue for the Gram matrix.** For the oracle inequality to hold, the Gram matrix has to satisfy a Restricted Eigenvalue (RE) property (condition (ii)), see [57]. For the linear Hawkes processes, in [28] the authors proved that the Gram matrix is invertible with high probability, which is stronger than a RE property. However, in a high dimensional framework, requiring the Gram matrix to be invertible may be too strong and too restrictive. Thus, as in [54] we turn instead to a RE property. Another possible, and similar, method is to introduce so called *random compatibility constants*, see [57], and [36] where this method is used in the framework of Hawkes estimation. However, neither [54] nor [36] prove, with high probability, that the RE property holds or that the random compatibility constants is positive. In [47], the authors prove a RE property with high probability for discrete time Hawkes processes using Kalikow decomposition, but, up to our knowledge there is not any similar result for heterogeneous processes. For HAR processes, we prove that the RE property holds with high probability, see Theorem 7.14. To achieve this, we use the coupling introduced in Section 6 in order to obtain concentration of the Gram matrix $G$ around its expectation $E[G]$ and finally, using a change of measure we link $E[G]$, which is not computable, to a quantity depending only on the chosen dictionary.

## 1.5. *Outline*

In Section 2 we rigorously define HAR processes and introduce the notations which will be used throughout the present article. Section 3 presents existence and stationary results. Linear HAR processes are introduced in Section 4 where we also provide the cluster representation. Exponential moments are derived in Section 5. In Section 6, we state the asymptotic limits, convergence to the stationary process and introduce the coupling and its properties leading to concentration
inequalities. In Section 7 we present the statistical estimation of the interaction functions using the LASSO. The proofs of results of Sections 3, 4, 5, 6 and 7 are given, respectively, in Sections 8, 9, 10, 11, 12. Finally, some technical results are presented in Appendices A, B, C, D, E.

**Notations.** We denote by $\mathbb{R}$ the set of real numbers and $\mathbb{Z}$ the set of the integers. By $\mathbb{R}_+$ we mean the non negative real numbers, ie $[0, \infty)$, and the non negative integers, $\{0, 1, 2, \dots\}$, are denoted by $\mathbb{N}$. We denote $\mathbb{N}^*$ (resp. $\mathbb{R}^*, \mathbb{R}_+^*$, etc) the set $\mathbb{N} \setminus \{0\}$ (resp. $\mathbb{R} \setminus \{0\}$, $\mathbb{R}_+ \setminus \{0\}$, etc).

## 2. Definition and Representation of HAR processes

Let $I \subset \mathbb{R}$. A simple point process $S$ on $I$ is a random set of distinct points lying in $I$ such that $\text{Card}(S \cap B)$ is measurable for any borel subset $B$ of $I$. $S$ is said to be *non exploding* if almost surely $S$ has only a finite number of points inside any bounded subset of $I$. The counting measure on $S$ is denoted by $dS$. For $C \subset I$ we denote by $S \cap C$ the set of points lying in $C$ and $S(C) := \text{Card}(S \cap C)$. A point process is fully determined by its counting measure or by the collection $(S(C))_{C \subset I}$. See [11] for more precise statements. In the sequel we will navigate freely between these different points of view.

Let $M$ a finite set. Consider a simple point process $S$ on $I$ with marks $Z \in M$, which means that for all $x \in S$, the mark $Z(x)$ is a random variable on $M$. For $m \in M$ we can define $S^m = \{x \in S | Z(x) = m\}$. $S$ is a $M$-multivariate point process on $I$ if $S^m$ is a point process for all $m \in M$ [11]. Note that by definition, $S^m$ and $S^{m'}$ have no common points for $m \neq m'$.

Let $M \in \mathbb{N}$ be the number of *nodes* and $P \in \mathbb{N}$ the number of *chains*. We denote by $M := \lfloor 1, M \rfloor$ the set of nodes and $P := \lfloor -P, -1 \rfloor$ the set of chains. Each chain $p \in P$ has a particular time embedding. More precisely the chain $p$ is a random function $W^p: \mathcal{D}_p \to \mathbb{R}$ where $\mathcal{D}_p := \theta_p + \frac{1}{n_p}\mathbb{Z}$. The positive integer $n_p \in \mathbb{N}^*$ is called the *frequency* of chain $p$ and the value $\theta_p \in [0, 1/n_p)$ is called the *phase* of chain $p$. We also denote by $\mathcal{D}^p$ the counting measure on $\mathcal{D}_p$ and $Q = \{(p, k), p \in P, k \in \mathcal{D}_p\}$.

Without loss of generality, we can suppose throughout the present article that the greatest common divisor of the frequencies $n_p$ for $p \in P$ is 1, ie

$$ (2.1) \qquad \gcd(n_p, p \in P) = 1. $$

In this article, to construct the various processes on the same underlying probability space we use the formalism of [19] to represent Hawkes processes as the solution to a system of SDEs driven by Poisson random measures. Ultimately this representation will ease couplings between HAR processes.

Let $(\Omega, \mathcal{F}, \mathbb{P})$ a probabilistic space with independent Poisson random measures $\pi^m$ on $\mathbb{R} \times \mathbb{R}_+$ for $m \in M$. Suppose that we also have independent random real variables $\xi_k^p$ for $(p, k) \in Q$ which are also independent of the Poisson random measures and such that for all $(p, k) \in Q$, for all $n \in \mathbb{Z}$, we have $\xi_k^p \sim \xi_{k+n}^p$ in distribution. Since almost surely each atom of all the Poisson random measures have distinct time coordinates, for simplicity assume that it is always the case. We define a filtration $(\mathcal{F}_t)_{t \in \mathbb{R}}$ by

$$ \mathcal{F}_t = \sigma \left( \pi^m \cap \left( (-\infty, t] \times \mathbb{R}_+ \right), m \in \mathcal{M} \text{ and } \xi_k^p, (p, k) \in Q, k \le t \right). $$

The collection of parameters associated to a HAR process is denoted $\mathfrak{M}$ and takes the following form

$$ \mathfrak{M} = ((\mu_m)_m, (h_{\beta}^{\alpha})_{\alpha, \beta}, (\Phi^{\alpha})_{\alpha}, (\kappa_p^{\alpha})_{\alpha, p}) $$

where we have for $\alpha, \beta \in M \cup P$, $m \in M$, $p \in P$

* A function $h_{\beta}^{\alpha}: \mathbb{R}_+^* \to \mathbb{R}$ called *interaction function of $\alpha$ on $\beta$*.

* A Lipschitz function $\kappa_p^{\alpha}: \mathbb{R} \to \mathbb{R}$ called *mask of chain $p$ on $\alpha$*.

* A non negative non decreasing Lipschitz function $\Phi^m: \mathbb{R} \to \mathbb{R}_+$ called *link function of $m$*.

* A Lipschitz function $\Phi^p: \mathbb{R} \to \mathbb{R}$ such that $\Phi^p(0) = 0$ called *link function of $p$*.

* A real constant $\mu_m$ called *base rate of node $m$*.
As for AR processes, the chains of HAR processes are constructed with some noises, the $(\xi_k^p)_{(p,k)\in Q}$. Since we do not require them to be centred, we adopt the terminology random drift which is less misleading than noise.

Parameters $\mathfrak{M}$ give birth to HAR processes $X = ((\lambda^m)_{m\in M}, (S^m)_{m\in M}, (W^p)_{p\in P})$ where the $(S^m)_{m\in M}$ is a multi-variate point process, $(\lambda^m)_{m\in M}$ are the predictable stochastic intensities of the point process, and $(W^p)_{p\in P}$, the chains, are discrete processes defined on $(D_p)_{p\in P}$. The process $X$ is defined as the solution of equations with parameters given by $\mathfrak{M}$. One can ask for the equations to be satisfied on $\mathbb{R}$ or just after a time $t_0 \in \mathbb{R}$. In this second case, the process is initialised before $t_0$ with an initial condition. An initial condition at time $t_0$ is a random variable $\mathcal{C}$ which is $\mathcal{F}_{t_0-}$ measurable and writes as follows

$$ \mathcal{C} = ((\mathcal{C}^m)_{m\in M}, (\mathcal{C}_k^p)_{(p,k)\in Q, k<t_0}), $$

where $\mathcal{C}^m$ is a subset of $(-\infty, t_0)$ that initialises the points of node $m$ before $t_0$. $\mathcal{C}_k^p$ belongs to $\mathbb{R} \cup \{\emptyset\}$ and initialises the chain $p$ at time $k$. If $\mathcal{C}_k^p$ takes the value $\emptyset$ we use the convention that for any function $\kappa_{\beta}^{\alpha}$ with $\alpha \in M \cup P$ and $\beta \in P$, we have $\kappa_{\beta}^{\alpha}(\emptyset) = 0$.

**Definition 2.1.** Let some parameters $\mathfrak{M}$. We say that a process $X = ((\lambda^m)_{m\in M}, (S^m)_{m\in M}, (W^p)_{p\in P})$, adapted to $(\mathcal{F}_t)_t$ and with $(\lambda^m)_{m\in M}$ predictable, is a $\mathbf{HAR}_{\mathfrak{M}}(-\infty)$ process, if we have

$$ (2.2a) \qquad \lambda_t^m = \Phi^m \left( \mu_m + \sum_{m' \in \mathcal{M}} \int_{-\infty}^{t-} h_{m'}^m (t-s) dS_s^{m'} + \sum_{p \in \mathcal{P}} \int_{-\infty}^{t-} h_p^m (t-k) \kappa_p^m (W_k^p) d\mathcal{D}_k^p \right) $$

$$ (2.2b) \qquad S^m(C) = \int_{C \times \mathbb{R}_+} 1_{x \le \lambda_s^m} d\pi^m(s, x) $$

$$ (2.2c) \qquad W_k^p = \xi_k^p + \Phi^p \left( \sum_{m \in \mathcal{M}} \int_{-\infty}^{k-} h_m^p (k-s) dS_s^m + \sum_{p' \in \mathcal{P}} \int_{-\infty}^{k-} h_{p'}^p (k-k') \kappa_{p'}^p (W_{k'}^p) d\mathcal{D}_{k'}^{p'} \right), $$

where (2.2a) holds $\mathbb{P}(d\omega) \otimes dt$ almost everywhere in $\Omega \times \mathbb{R}$, (2.2b) holds almost surely for all $C \subset \mathbb{R}$ and (2.2c) holds almost surely for all $(p, k) \in Q$. Such a process is called a HAR process with parameters $\mathfrak{M}$ started at $-\infty$.

Let $t_0 \in \mathbb{R}$ and an initial condition $\mathcal{C}$ at time $t_0$. We say that the process $X$, adapted to $(\mathcal{F}_t)_t$ and with $(\lambda^m)_{m\in M}$ predictable, is a $\mathbf{HAR}_{\mathfrak{M}}(t_0, \mathcal{C})$ process, if

1. we have $S^m \cap (-\infty, t_0) = \mathcal{C}^m$ for all $m \in \mathcal{M}$ and $W_k^p = \mathcal{C}_k^p$ for all $(p, k) \in Q$ with $k < t_0$,

2. (2.2a) holds $d\mathbb{P}(\omega) \otimes dt$ almost everywhere in $\Omega \times [t_0, \infty)$, (2.2b) holds almost surely for all $C \subset [t_0, \infty)$ and (2.2c) holds almost surely for all $(p, k) \in Q$ with $k \ge t_0$.

Such a process is called a HAR process with parameters $\mathfrak{M}$ started at $t_0$ with initial condition $\mathcal{C}$.

Some remarks are in order.

The realisation of point processes by SDEs driven by Poisson random measures is done in (2.2b). The intensity process, $\lambda_t^m$, is defined by (2.2a) and then $S^m$ is defined to be atoms of the Poisson random measures lying below the intensity, which is the classical thinning for Hawkes processes [19, 48]. As shown in Appendix A, $\lambda$ is indeed the intensity of $S$, and, whenever $\lambda$ is defined $d\mathbb{P}(\omega) \otimes dt$ almost everywhere, $S$ is well defined.

We allow $\mathcal{C}_k^p$ to take the value $\emptyset$ for the following reason. One may want to have a HAR process started at time $t_0$ without initial condition, thus the logical choice is $\mathcal{C}^m = \emptyset$ and $\mathcal{C}_k^p = 0$. However, since the masks may have $\kappa_{\beta}^{\alpha}(0) \ne 0$, this choice of initial condition is not the desired one. Allowing the value $\emptyset$ and forcing $\kappa_{\beta}^{\alpha}(\emptyset) = 0$ for any $\alpha, \beta$ creates the desired initial condition.

For simplicity, we only write $\mathbf{HAR}_{\mathfrak{M}}(t_0)$ when the initial condition $\mathcal{C}$ is chosen to be the empty one ($\mathcal{C}^m = \emptyset$ for all $m$ and $\mathcal{C}_k^p = \emptyset$ for all $(p, k) \in Q$ with $k < t_0$) and we denote this initial condition by $\mathcal{C}^{\mathfrak{M}}$. In practice, it is equivalent to replace $-\infty$ by $t_0$ in the lower bounds of the integrals in (2.2a), (2.2b) and (2.2c).

Observe that if there are no chains, meaning $P = \emptyset$, we obtain the classical Hawkes model. If there is no nodes, meaning $M = \emptyset$, the chains satisfy the multivariate and multiscale version of (1.3).
The predictable intensity of the point process $S^m$ associated to node $m$ is $\lambda^m$. It satisfies (2.2a). Due to the function $\Phi^m$, $\lambda^m$ is nonlinear with respect to the parameters of the model. Thus the HAR processes we consider are very general. The linear quantity

$$ (2.3) \qquad \ell_t^m := \mu_m + \sum_{m' \in M} \int_{-\infty}^{t-} h_{m'}^m(t-s) dS_s^{m'} + \sum_{p \in P} \int_{-\infty}^{t-} h_p^m(t-k) \kappa_p^m (W_k^p) d\mathcal{D}_k^p, $$

will be called *pre-intensity* in this article. It is defined so that $\lambda_t^m = \Phi^m(\ell_t^m)$. It is made of three quantities being from left to right: a base rate (as in Hawkes processes), interactions from nodes on node $m$ (as in Hawkes processes) and interactions from chains on node $m$. The pre-intensity $\ell^m$ can take negative values, but since $\Phi^m \ge 0$ the intensity $\lambda^m$ is well defined.

If the parameters $\mu_m$, $h_{m'}^m$, $h_p^m$, $\kappa_p^m$ for $m' \in M$ and $p \in P$ are non negative, then we have $\ell_t^m \ge 0$. Thus, in this case one can take $\Phi^m = \max(0, \cdot)$ and has $\lambda^m = \ell^m$. It is morally the same thing as taking $\Phi^m = \text{id}$. And one obtains, if $P = \emptyset$, the well known linear Hawkes process.

Chains satisfy (2.2c). As for classical AR processes, this equation is made of two elements: a possibly non centered noise $\xi_k^p$ called random drift, and a term that depends on the past. The link function $\Phi^p$ is required to satisfy $\Phi^p(0) = 0$. It is not restrictive since a situation where $\Phi^p(0) \ne 0$ fits our case by doing these modifications: $(\Phi^p, \xi^p) \leftarrow (\Phi^p - \Phi^p(0), \xi^p + \Phi^p(0))$. As for the equation of the intensity, the link function $\Phi^p$ allows the general nonlinear case. Inside the link function, we have the interactions from the nodes on chain $p$ and then the classical AR term of interactions from chains on chain $p$.

Let us introduce some notations.

The Lipschitz coefficients of the masks and the link functions are denoted as follows

$$ (2.4) \qquad a_p^\alpha := \text{Lip}(\kappa_p^\alpha), \quad L_\alpha := \text{Lip}(\Phi^\alpha) \quad p \in P, \alpha \in M \cup P. $$

For a function $f: \mathbb{R} \to \mathbb{R}$ and a positive real number $\alpha > 0$ we define the following norm which takes into account discrete behaviour at frequency scale $\alpha$,

$$ \|f\|_{1,\alpha}^{\infty} := \sup_{x \in \mathbb{R}} \sum_{k \in \mathbb{Z}} |f(x + k/\alpha)|. $$

In particular, if $f$ is supported on $\mathbb{R}_+^*$, then since $D_p = \theta_p + \frac{1}{n_p}\mathbb{Z}$ we have

$$ \|f\|_{1,n_p}^{\infty} = \sup_{t \in \mathbb{R}} \int_{-\infty}^{t-} |f(t-k)| d\mathcal{D}_k^p. $$

The support of a function $f: I \to \mathbb{R}$ with $I \subset \mathbb{R}$, is denoted by $\text{supp}(f)$.

A real function $f$ is locally integrable, i.e. $f \in L_{\text{loc}}^1$, if for all compact sets $K \subset \mathbb{R}$, we have

$$ \int_K |f(x)| dx < \infty. $$

The identity matrix is denoted by $\text{Id}_n$ with $n$ the dimension or $\text{Id}$ if there is no ambiguity. For a (complex valued) square matrix $N$ the spectral radius of $N$, defined as the supremum of the magnitudes of the eigenvalues of $N$, is denoted as $\text{SpR}(N)$.

For $q \ge 1$, the $\ell^q$ norm of the vector $x \in \mathbb{R}^n$ is denoted by $|x|_q$,

$$ |x|_q = \left( \sum_{i=1}^{n} |x_i|^q \right)^{1/q}. $$

Given two vector $x$ and $y$ (or matrices) of same dimension, we use $\leq$ for the entry by entry comparison,

$$ x \leq y \iff \forall i, x_i \leq y_i. $$

Finally, for a vector $x$ and a real function $f$, we denote by $f(x)$ the coordinate wise application of $f$.
### 3. Existence results for HAR processes

Equations defining HAR processes are implicit, both the point processes and the chains depend on each others. Thus it is not clear that HAR processes exist. In this section we derive conditions to ensure existence of HAR processes. In case of Hawkes processes, it is well known whether or not there exist solutions to the equation defining Hawkes processes [3, 12]. The main difference here is that the Hawkes process is mixed with discrete sequences driven by random drifts. For HAR processes started at a time $t_0 \in \mathbb{R}$ with suitable initial condition (see Definition 3.2), since the present value of the process depends only on the past we can construct the process on increasing intervals $[t_0, t_0 + t]$ with $t \to \infty$, see Theorem 3.4. However for HAR started at $t = -\infty$, see Theorem 3.9, this constructive proof of existence is not relevant and we use instead a fixed point method similar to the one implemented in [12].

#### 3.1. Existence of non exploding HAR processes with initial condition

In this subsection we present the results on existence of HAR processes started at a finite time with suitable initial condition. We are only interested in processes that do not blow up to infinity in finite time: we need them to be non exploding, which means that the process is bounded on compact sets. This is explained by the following definition.

**Definition 3.1 (Non exploding process).** A process $((\lambda^m)_{m \in M}, (S^m)_{m \in M}, (W^p)_{p \in P})$ is said to be non exploding if

1. $\forall m \in M, \forall C \subset \mathbb{R} \text{ bounded, } S^m(C) < \infty$,

2. $\forall m \in M, \forall t \in \mathbb{R}, |\lambda_t^m| < \infty$,

3. $\forall p \in P, \forall k \in D_p, |W_k^p| < \infty$.

For a HAR process to be non exploding, one cannot allow arbitrary initial conditions. The following definition introduces integrable initial conditions.

**Definition 3.2 (Integrable initial condition).** An initial condition $\mathcal{C}$ at time $t_0 \in \mathbb{R}$ is said integrable if there exists a constant $K < \infty$ such that

* for all $m \in M$, for all intervals $B \subset \mathbb{R}$ of length 1, $E[\text{Card}(\mathcal{C}^m \cap B)] \le K$,

* for all $(p, k) \in \mathcal{Q}$, we have $E[1_{\mathcal{C}_k^p \ne \emptyset} | \mathcal{C}_k^p] \le K$.

Since we are interested in existence and uniqueness results, we need to introduce the right notion of uniqueness for HAR processes.

**Definition 3.3 (Modification).** Two processes $((\lambda^m)_{m \in M}, (S^m)_{m \in M}, (W^p)_{p \in P})$ and $((\tilde{\lambda}^m)_{m \in M}, (\tilde{S}^m)_{m \in M}, (\tilde{W}^p)_{p \in P})$ are said to be modifications of each others if

* For all $m \in M$, $\lambda_t^m = \tilde{\lambda}_t^m$ almost everywhere in $\Omega \times \mathbb{R}$ for the measure $d\mathbb{P}(\omega) \otimes dt$.

* Almost surely, for all $m \in M$, $S^m = \tilde{S}^m$.

* Almost surely, for all $(p, k) \in \mathcal{Q}, W_k^p = \tilde{W}_k^p$.

Let us introduce the following assumption.

**Assumption (NoExpl).** Parameters $\mathfrak{M}$ satisfy assumption (NoExpl) if for all $m \in M$ the link function $\Phi^m$ is Lipschitz, i.e. $L_m < \infty$, and for all $m, m' \in M$, for all $p \in P$ the functions $h_{m'}^m, h_m^m$ and $h_m^{p'}$ are in $L^1$.

We can now state the main result of the paragraph.

**Theorem 3.4 (Existence of non exploding HAR processes).** Let some parameters $\mathfrak{M}$ and an initial condition $\mathcal{C}$ at time $t_0$. Suppose that assumption (NoExpl) $[\mathfrak{M}]$ holds and that $\mathcal{C}$ is integrable. Then there exists a unique (up to modification) HAR$_{\mathfrak{M}}(t_0, \mathcal{C})$ process among the ones which are non exploding.

The assumptions of Theorem 3.4 are very mild and general and the initial condition can be taken as Poisson point processes, or even some kind of sufficiently evenly spaced deterministic points. For the chains, one can take iid random variables with finite expectation for example. However, if one wants to relax the $L^1$ assumptions into only $L_{\text{loc}}^1$ assumptions it is possible at the cost of other conditions on the initial condition, as explained in the following remark.

**Remark 3.5.** An initial condition $\mathcal{C}$ at time $t_0 \in \mathbb{R}$ is said finite if

* almost surely, for all $m \in M$, $\text{Card}(\mathcal{C}^m) < \infty$,
• almost surely, we have

$$ \sup \{t < t_0 \mid \forall (p, k) \in \mathcal{Q} \text{ with } k \le t, \mathcal{C}_k^p = \emptyset \} > -\infty, $$

with the convention that $\sup(\emptyset) = -\infty$.

Finite initial conditions are not necessarily integrable and vice versa.

**Assumption (NoExplBis).** Parameters $\mathfrak{M}$ satisfy assumption (NoExplBis) if for all $m \in \mathcal{M}$ the link function $\Phi^m$ is Lipschitz, ie $L_m < \infty$, and for all $m, m' \in \mathcal{M}$, for all $p \in \mathcal{P}$ the functions $h_{m'}^m, h_p^m$ are in $L_{\text{loc}}^1$.

Theorem 3.4 applies to finite initial conditions under assumption (NoExplBis) [M]. The proof is almost exactly the same as explained in Remark 8.1. Assumption (NoExplBis) [M] matches the classical criterion of non explosion of linear Hawkes processes with empty initial condition: the interacting functions are $L_{\text{loc}}^1$, see [3]. This criterion naturally extends to nonlinear Hawkes process by assuming in addition that the link functions, here the $\Phi^m$, are Lipschitz.

In the sequel, we keep assumption (NoExpl) and integrable initial conditions.

## 3.2. Existence of stationary HAR processes

Establishing existence of stationary processes is crucial in particular for practical purposes, for example in statistical applications [28], or because they appear as long time limits of non stationary processes [12]. In general, HAR processes started at a finite time are not stationary. This motivates to look at HAR processes started at $-\infty$. Since HAR processes are a mix between discrete and continuous processes, we need to define what stationary means for HAR processes.

**Definition 3.6 (1-stationary).** A process $X = ([\lambda_t^m]_{t \in \mathbb{R}}, m \in \mathcal{M}], [S^m, m \in \mathcal{M}], [(W_k^p]_{k \in \mathcal{D}_p}, p \in \mathcal{P}])$ is said to be 1-stationary if

$$ X \stackrel{\text{law}}{=} ([\lambda_{t+1}^m]_{t \in \mathbb{R}}, m \in \mathcal{M}], [S^m + 1, m \in \mathcal{M}], [(W_{k+1}^p]_{k \in \mathcal{D}_p}, p \in \mathcal{P}]) $$

**Remark 3.7.** Stationarity is defined with a period equal to 1. We cannot decrease the period since we assume that the greatest common divisor of the $n_p$ for $p \in \mathcal{P}$ is 1, see (2.1). Thus the family of time indexes of the chains, meaning $(\mathcal{D}_p)_{p \in \mathcal{P}}$, is only 1-periodic: $(\mathcal{D}_p)_{p \in \mathcal{P}} = (\mathcal{D}_p + 1)_{p \in \mathcal{P}}$.

The classical condition to prove the existence of a stationary Hawkes process is based on the spectral radius of a matrix which captures the interactions between nodes. More precisely, the spectral radius of the matrix containing the $L^1$ norms of the interaction functions has to be strictly smaller than one [12]. Our condition will be of the same kind. The following definition introduces the matrix of the norms of the interactions between both nodes and chains on which the major assumption to the existence of 1-stationary HAR processes relies.

**Definition 3.8.** Let $\mathfrak{M}$ some parameters. Recall that for $m \in \mathcal{M}, p, p' \in \mathcal{P}$

$$ a_{p'}^p = \text{Lip}(\kappa_{p'}^p), \quad a_p^m = \text{Lip}(\kappa_p^m), \quad L_m = \text{Lip}(\Phi^m), \quad L_p = \text{Lip}(\Phi^p). $$

We introduce the following matrices:

$$ H_S^S = (L_m \|h_{m'}^m\|_1)_{m,m' \in \mathcal{M}}, \quad H_W^S = (L_m a_p^m \|h_p^m\|_{1,n_p}^\infty)_{m \in \mathcal{M}, p \in \mathcal{P}}, \quad H_S^W = (L_p \|h_m^p\|_1)_{p \in \mathcal{P}, m \in \mathcal{M}} $$

$$ \text{and } H_W^W = (L_p a_{p'}^p \|h_{p'}^p\|_{1,n_{p'}}^\infty)_{p, p' \in \mathcal{P}}. $$

Finally, the matrix $H_{\mathfrak{M}}$ is defined by

$$ H_{\mathfrak{M}} = \begin{pmatrix} H_S^S & H_W^S \\ H_S^W & H_W^W \end{pmatrix}. $$

Infinite values are allowed with the convention that $\infty \times 0 = 0 \times \infty = \infty$.

Let us introduce some assumptions before stating the theorem about 1-stationary HAR processes.

**Assumption (Spec1).** We say that parameters $\mathfrak{M}$ satisfy assumption (Spec1) if the spectral radius of the matrix $H_{\mathfrak{M}}$ introduced in Definition 3.8 is smaller than one, equivalently $\text{SpR}(H_{\mathfrak{M}}) < 1$.
**Assumption (FiMe).** The random drifts have finite mean, equivalently,

$$ \sup_{(p,k) \in \mathcal{Q}} \mathbb{E}[|\xi_k^p|] < \infty. $$

Now, we can state the main result concerning existence of HAR processes started at $-\infty$. It also answers questions about uniqueness and 1-stationary properties.

**Theorem 3.9 (Stationary HAR processes).** Let some parameters $\mathfrak{M}$. Suppose that assumptions (Spec1)[$\mathfrak{M}$] and (FiMe) hold. Then, up to modification, there exists a unique $\mathbf{HAR}_{\mathfrak{M}}(-\infty)$ process $((\lambda^m)_{m \in \mathcal{M}}, (S^m)_{m \in \mathcal{M}}, (W^p)_{p \in \mathcal{P}})$ such that

$$ \sup_{m \in \mathcal{M}, t \in \mathbb{R}} \mathbb{E}[|\lambda_t^m|] < \infty \quad \text{and} \quad \sup_{p \in \mathcal{P}, k \in \mathcal{D}_p} \mathbb{E}[|W_k^p|] < \infty. $$

Moreover, this process is also 1-stationary.

For classical Hawkes processes, Brémaud and Massoulié in [12] have proved that a stationary version exists if the spectral radius of the matrix gathering the $L^1$ norms of the interaction functions is smaller than one. Assumption (Spec1)[$\mathfrak{M}$] naturally extends this condition. Assumption (FiMe) is a very mild assumption on the random drifts. In addition of providing an existence result, Theorem 3.9 also shows uniqueness among processes bounded in $L^1$. Finally, it validates the intuition that a process started at $-\infty$ is stationary since it has already evolved during an infinite period of time.

## 4. Linear HAR processes and cluster representation

In the previous section we established existence results, to continue our study we aim for a specific class of HAR processes on which a lot of calculations are possible. To draw a parallel, in the study of Hawkes processes, linear Hawkes processes play a major role since they allow lot of calculations due to their famous cluster representation [31]. Cluster representation allows to quantify exactly and isolate the impact of each point in the future. We will be able to derive a similar cluster representation for HAR processes with linear parameters. We define the $\mathcal{L}$ class, the class of linear parameters, as follows.

**Definition 4.1 (L class).** A set a parameters $\mathfrak{M}$ is said to be linear, or equivalently $\mathfrak{M} \in \mathcal{L}$ if

* (linearity 1) $\forall m \in \mathcal{M}, \Phi^m = \text{id}$ and $\forall p \in \mathcal{P}, \Phi^p = \text{id}$,

* (linearity 2) $\forall \alpha \in \mathcal{M} \cup \mathcal{P}, \forall p \in \mathcal{P}$ we have $\kappa_p^\alpha(x) = a_p^\alpha x + b_p^\alpha$ for any $x \in \mathbb{R}$ and $\kappa_p^\alpha(\emptyset) = 0$, with $a_p^\alpha, b_p^\alpha$ non negative constants,

* (positiveness) $\forall \alpha, \beta \in \mathcal{M} \cup \mathcal{P}$, we have $h_\beta^\alpha \ge 0$ and $\forall m \in \mathcal{M}, \mu_m \ge 0$.

**Remark 4.2.** To ensure that the intensity remains positive, if there is an initial condition it has to be non negative (ie $\mathcal{E}_k^p \ge 0$ or $\mathcal{E}_k^p = \emptyset$). The random drifts $\xi_k^p$ also have to be non negative. The definition of $\kappa_p^\alpha$ remains coherent with $\text{Lip}(\kappa_p^\alpha) = a_p^\alpha$.

There are two major linearity assumptions defining the $\mathcal{L}$ class, a first one on the link functions and a second one on the masks. Positiveness assumption, in addition with non negative random drifts $\xi_k^p$, is here to ensure that in the end the intensity is non negative. Recall the equation of the chains (2.2c),

$$ W_k^p = \xi_k^p + \Phi^p \left( \sum_{m \in \mathcal{M}} \int_{-\infty}^{k-} h_m^p(k-s) dS_m^m + \sum_{p' \in \mathcal{P}} \int_{-\infty}^{k-} h_{p'}^p(k-k') \kappa_{p'}^p (W_{k'}^{p'}) d\mathcal{D}_{k'}^{p'} \right). $$

Under the assumptions on the masks and the link functions, this equation becomes a linear relation on the past values of chains. One has, heuristically and in the one dimensional case,

$$ \begin{aligned} W_k &= \xi_k + (\text{points linear contribution up to } k) + (\text{non random term}) + \sum_{k' < k} ah(k - k')W_{k'} \\ &= (\text{something that depends on } k) + \sum_{k' < k} ah(k - k')W_{k'}. \end{aligned} $$

This recurrence relation is the starting point to derive the cluster representation for HAR processes. By solving it completely we will be able to express the chains only in terms of linear functions of points and random drifts. Then introducing this expression of the chains in the intensity equation will lead us to the cluster representation. In the next subsections we solve completely this type of recurrence relation to then be able to apply these results to HAR processes.
### 4.1. Recurrence relation of chains

The main assumptions in the $\mathcal{L}$ class are (linearity 1) and (linearity 2) on the link functions and the masks. In the general case each chain depends on the previous ones through a complex recurrence relationship. What makes these two assumptions interesting is that the complex dependence of a chain on the previous ones becomes a linear dependence! Let us give a simple example. Let a sequence $(w_n)_{n \ge 0}$ defined as follows

$$w_n = u_n + \sum_{0 \le k < n} h(n-k)w_k$$

where the sequence $u$ is given and $h$ is a fixed (say non negative) function. A natural wish is to have an expression of $w_n$ depending only on the sequence $u$. To do so, we need to unroll the recurrence relationship. In the end, it is easy to be convinced that $w_n$ can be expressed as follows

$$w_n = \sum_{0 \le k \le n} C_k^n u_k$$

with $(C_k^n)_{n \ge k}$ be some coefficients depending only on $h$, see classical AR theory [10, 42, 53]. These coefficients grasp the complexity of the nested dependence of the sequence $w$ on itself. One can prove for instance that these coefficients satisfy

* For $k > k'$ we have

$$C_{k'}^k = \sum_{k' \le k'' < k} h(k - k'') C_{k'}^{k''}$$

* For all $k \ge k'$, $C_{k'+1}^{k+1} = C_k^k$

* If $\sum_{k \ge 0} h(k) =: \alpha < 1$ then

$$\sum_{k=-\infty}^{n} C_k^n < \frac{1}{1-\alpha}.$$

We extend these results to the multidimensional case (since we have $P$ chains) and on a domain $(\mathcal{D}_p)_{p \in \mathcal{P}}$, more complex than the integers.

The following lemma introduces the coefficients $\langle \begin{bmatrix} p, k \\ p', k' \end{bmatrix} \rangle_{\mathfrak{M}}$ which are the analogs of the $C_{k'}^k$'s in the multidimensional case.

**Lemma 4.3.** Let parameters $\mathfrak{M} \in \mathcal{L}$. There exists a unique set of coefficients $(\langle \begin{bmatrix} p, k \\ p', k' \end{bmatrix} \rangle_{\mathfrak{M}})_{(p,k), (p',k') \in \mathcal{Q}, k \ge k'}$ such that,

1. For any $(p_0, k_0) \in \mathcal{Q}$, for any sequence $(Z_k^p)_{(p,k) \in \mathcal{Q}, k \ge k_0}$, the sequence $(V_k^p)_{(p,k) \in \mathcal{Q}, k \ge k_0}$ defined by

$$V_k^p = Z_k^p + \sum_{p' \in \mathcal{P}} \int_{k_0}^{k-} a_{p'}^p h_{p'}^p (k-k') V_{k'}^{p'} d\mathcal{D}_{k'}^{p'} \quad (4.1)$$

satisfies the following equality,

$$V_k^p = \sum_{p' \in \mathcal{P}} \int_{k_0}^{k} \langle \begin{bmatrix} p, k \\ p', k' \end{bmatrix} \rangle_{\mathfrak{M}} Z_{k'}^{p'} d\mathcal{D}_{k'}^{p'}.$$

2. The coefficients $(\langle \begin{bmatrix} p, k \\ p', k' \end{bmatrix} \rangle_{\mathfrak{M}})_{(p,k), (p',k') \in \mathcal{Q}, k \ge k'}$ are 1-periodic, which means that for every $n \in \mathbb{Z}$,

$$\langle \begin{bmatrix} p, k+n \\ p', k'+n \end{bmatrix} \rangle_{\mathfrak{M}} = \langle \begin{bmatrix} p, k \\ p', k' \end{bmatrix} \rangle_{\mathfrak{M}}.$$
3. For all $(p_0, k_0) \in \mathcal{Q}$ we have $\langle p_0, k_0 \rangle_{\mathfrak{M}} = 1$. If there exists $(p_1, k_1) \in \mathcal{Q}$ with $k_1 = k_0$ then $\langle p_1, k_0 \rangle_{\mathfrak{M}} = 1_{p_0=p_1}$. Finally for $(p', k') \in \mathcal{Q}$ with $k > k'$,

$$ \langle p, k \rangle_{\mathfrak{M}} = \sum_{p'' \in \mathcal{P}} \int_{k'}^{k-} a_{p''}^p h_{p''}^p (k-k'') \langle p'', k'' \rangle_{\mathfrak{M}} d\mathcal{D}_{k''}^{p''} \geq 0. $$

Lemma 4.3 is about the combinatorial properties of the coefficients. The coefficients only depend on $\mathfrak{M}$, they can be computed recursively with the third point of Lemma 4.3. They are encoding all the dynamic of the recurrence relation defined by the functions $a_{p'}^p h_{p'}^p(\cdot)$.

Recall from Definition 3.8 that $H_W^W$ is the matrix $(L_p a_{p'}^p \|h_{p'}^p\|_{1,n_{p'}}^{\infty})_{p,p' \in \mathcal{P}}$. We recall that for linear parameters $L_p = 1$. Let the following assumption.

**Assumption (Spec1.1).** WZ say that parameters $\mathfrak{M}$ satisfy assumption (Spec1.1) if the spectral radius of the matrix $H_W^W$ defined in Definition 3.8 is smaller than one, equivalently $\text{SpR}(H_W^W) < 1$.

Under assumption (Spec1.1) we obtain analytic results on the coefficients, see Lemma B.1 in the Appendix B. These results are useful in the sequel, for example to obtain controls on clusters.

We end this subsection with the following result that gives a closed form for the coefficients.

Let $(p, k) \in \mathcal{Q}$ and $(p', k') \in \mathcal{Q}$ with $k' < k$. A path $\gamma$ from $(p', k')$ to $(p, k)$ is a sequence $(p_1, k_1), \dots, (p_n, k_n)$ in $\mathcal{Q}$, where $n \ge 2$ is an integer and such that $(p_1, k_1) = (p', k')$, $(p_n, k_n) = (p, k)$ and $k_1 < \dots < k_n$. Given parameters $\mathfrak{M} \in \mathcal{L}$ and a path $\gamma = (p_1, k_1), \dots, (p_n, k_n)$ from $(p', k')$ to $(p, k)$ we denote

$$ Y_{\mathfrak{M}}(\gamma) := \prod_{i=1}^{n-1} a_{p_i}^{p_{i+1}} h_{p_i}^{p_{i+1}}(k_{i+1} - k_i). $$

**Proposition 4.4 (Closed form for coefficients).** Let some parameters $\mathfrak{M} \in \mathcal{L}$. Let $(p', k') \in \mathcal{Q}$ and $(p, k) \in \mathcal{Q}$ with $k' < k$. Then we have

$$ \langle p, k \rangle_{\mathfrak{M}} = \sum_{\gamma: (p', k') \to (p, k)} Y_{\mathfrak{M}}(\gamma). $$

## 4.2. Cluster representation for linear HAR processes

In this subsection we use the coefficients defined in the previous subsection to unroll the nested dynamic of chains and reveal the cluster representation for linear HAR processes. Let parameters $\mathfrak{M} \in \mathcal{L}$ and $t_0 \in \{-\infty\} \cup \mathbb{R}$. We state the result for HAR processes without initial condition. As explain in Remark 4.6, a similar result can be written for HAR processes with initial condition with the exact same ideas up to more complex writing.

Let us define quantities that are useful to express the chains in a non nested way. Let $(p, k) \in \mathcal{Q}$ with $k \ge t_0$ and define,

$$ (4.2) \qquad \left\{ \begin{array}{l} Y_{p,k}^b(t_0) = \sum_{p' \in \mathcal{P}} \int_{t_0}^k \langle p, k \rangle_{\mathfrak{M}} \left[ \sum_{p'' \in \mathcal{P}} b_{p''}^{p'} \int_{t_0}^{k'} h_{p''}^{p'} (k' - k'') d\mathcal{D}_{k''}^{p''} \right] d\mathcal{D}_{k'}^{p'} \\ Y_{p,k}^{\xi}(t_0) = \sum_{p' \in \mathcal{P}} \int_{t_0}^k \langle p, k \rangle_{\mathfrak{M}} \xi_{k'}^{p'} d\mathcal{D}_{k'}^{p'} \\ Y_{p,k}^m(t_0, t) = \sum_{p' \in \mathcal{P}} \int_{t_0}^k \langle p, k \rangle_{\mathfrak{M}} \mathbb{1}_{\{t < k'\}} h_m^{p'}(k' - t) d\mathcal{D}_{k'}^{p'}. \end{array} \right. $$

Each of these quantities encloses the total impact at time $k$ on chain $p$ of, from up to down, the constants $b$'s of the masks, the random drifts, node $m$. If $t_0 = -\infty$ we may omit $t_0$ in the notation an only write only $Y_{p,k}^b$ and so on.

The following result is the main theorem of this section, it states a cluster representation for linear models.
**Theorem 4.5** (Cluster representation for linear HAR processes). Let $\mathfrak{M} \in \mathcal{L}$. If $t_0 \in \mathbb{R}$ suppose that assumption (NoExpl) $[\mathfrak{M}]$ holds and if $t_0 = -\infty$ suppose that assumptions (Spec1) $[\mathfrak{M}]$ and (FiMe) hold. Suppose also that the random drifts are non negative, ie $\xi_k^p \ge 0$ for all $(p, k) \in \mathcal{Q}$. Then the $\mathbf{HAR}_{\mathfrak{M}}(t_0)$ process from Theorem 3.4 (if $t_0 \in \mathbb{R}$) or Theorem 3.9 (if $t_0 = -\infty$) must be given by

$$W_k^p = Y_{p,k}^b(t_0) + Y_{p,k}^\xi(t_0) + \sum_m \int_{t_0}^{k-} Y_{p,k}^m(t_0, t) dS_t^m, \quad (p, k) \in \mathcal{Q}, k \ge t_0,$$

and there exists a cluster representation given by

$$\lambda_t^m = \mathcal{I}^m(t_0, t) + \sum_{m'} \int_{t_0}^{t-} \mathfrak{h}_{m'}^m(s, t) dS_s^{m'}, \quad m, m' \in \mathcal{M}, t \ge t_0,$$

where any point, say from node $m_0$ at time $s \in \mathbb{R}$, has an impact on the intensity at time $t > s$ of node $m$ being

$$(4.3) \qquad \mathfrak{h}_{m_0}^m(s, t) = h_{m_0}^m(t-s) + \sum_{p, p' \in \mathcal{P}} \int_{s^+}^{t-} \int_{s^+}^{k} a_p^m h_p^m(t-k) \langle \begin{bmatrix} p, k \\ p', k' \end{bmatrix} \rangle_{\mathfrak{M}} h_{m_0}^{p'}(k' - s) d\mathcal{D}_{k'}^{p'} d\mathcal{D}_k^p,$$

and the immigrant rate is given by

$$(4.4) \qquad \mathcal{I}^m(t_0, t) = \mu_m + \sum_{p \in \mathcal{P}} \int_{t_0}^{t-} \left[ h_p^m(t-k) \times \left( a_p^m Y_{p,k}^b(t_0) + a_p^m Y_{p,k}^\xi(t_0) + b_p^m \right) \right] d\mathcal{D}_k^p.$$

Finally if assumption (Spec1.1) $[\mathfrak{M}]$ holds (which is the case under assumption (Spec1) $[\mathfrak{M}]$),

$$(4.5) \qquad \|\mathfrak{h}_{m_0}^m(s, \cdot)\|_{1, (s, \infty)} \le \|h_{m_0}^m\|_1 + \sum_{p, p' \in \mathcal{P}} n_p a_p^m \|h_p^m\|_1 \left[ (\mathrm{Id} - H_W^W)^{-1} \right]_{p, p'} \|h_{m_0}^{p'}\|_{1, 1}^\infty.$$

Each immigrant point (coming from the immigration part of the intensity) is a new root for a new cluster. Indeed, each immigrant point, say from node $m$ at time $s$, will have some children since it impacts the future intensity of node $m'$ at time $s+t$ by $\mathfrak{h}_m^{m'}(s, s+t)$. Similarly, each child will have children and so on, creating a cluster, ie the whole family of the initial immigrant point. Conditionally to the immigrants, all the clusters are independent. See Figure 1. Cluster representation is a powerful tool, to derive result on the whole process; one can look at what happens for a single cluster and then the it only remains to superpose the contributions of each clusters, which is usually fine since the clusters are independent.

<figure><img src="image_3.png" /><figcaption>FIG 1. Illustration of the clusters. The red dots are the immigrant points. They arrive with the immigrant rate and give birth to clusters, represented by the blue shapes.</figcaption></figure>
**Remark 4.6.** Let $t_0 \in \mathbb{R}$ and $\mathcal{C}$ a non negative integrable initial condition at time $t_0$, which means that $\mathcal{C}_k^p \ge 0$ or $\mathcal{C}_k^p = \emptyset$ for all $(p, k) \in \mathcal{Q}$ with $k < t_0$. Let

$$ (4.6) \qquad \left\{ \begin{aligned} J_t^{m, \mathcal{C}}(t_0) &= \sum_{m' \in \mathcal{M}} \int_{-\infty}^{t_0-} h_{m'}^m(t-s) d\mathcal{C}_s^{m'} + \sum_{p \in \mathcal{P}} \int_{-\infty}^{t_0-} h_p^m(t-k) \kappa_p^m(\mathcal{C}_k^p) d\mathcal{D}_k^p \\ J_k^{p, \mathcal{C}}(t_0) &= \sum_{m \in \mathcal{M}} \int_{-\infty}^{t_0-} h_m^p(k-s) d\mathcal{C}_s^m + \sum_{p' \in \mathcal{P}} \int_{-\infty}^{t_0-} h_{p'}^p(k-k') \kappa_p^m(\mathcal{C}_{k'}^{p'}) d\mathcal{D}_{k'}^{p'} \\ Y_{p,k}^{\mathcal{C}}(t_0) &= \sum_{p' \in \mathcal{P}} \int_{t_0}^{k} \langle p, k \rangle_{\mathfrak{M}} J_{k'}^{p', \mathcal{C}}(t_0) d\mathcal{D}_{k'}^{p'} \end{aligned} \right. $$

$J_t^{m, \mathcal{C}}(t_0)$ and $J_k^{p, \mathcal{C}}(t_0)$ correspond, respectively, to the terms in (2.2a) and (2.2c) coming from the initial condition. $Y_{p,k}^{\mathcal{C}}(t_0)$ correspond to the resulting term coming from the initial condition in $W_k^p$ after solving the nested recurrence relation of the chains. Then, if $\mathcal{C}$ is integrable, under the assumptions of Theorem 4.5, up to the following underlined changes, the result of Theorem 4.5 is the same;

$$ W_k^p = \underline{Y_{p,k}^{\mathcal{C}}(t_0)} + Y_{p,k}^b(t_0) + Y_{p,k}^{\xi}(t_0) + \sum_m \int_{t_0}^{k-} Y_{p,k}^m(t_0, t) dS_t^m, $$

$$ \mathcal{I}^m(t_0, t) = \mu_m + \frac{J_t^{m, \mathcal{C}}(t_0)}{\sum_{p \in \mathcal{P}} \int_{t_0}^{t-} \left[ h_p^m(t-k) \times \left( a_p^m \underline{Y_{p,k}^{\mathcal{C}}(t_0)} + a_p^m Y_{p,k}^b(t_0) + a_p^m Y_{p,k}^{\xi}(t_0) + b_p^m \right) \right] d\mathcal{D}_{k'}^p d\mathcal{D}_k^p. $$

### 4.3. The structure of clusters

In this subsection we dive into the study of clusters. The following definition recalls the expression of the functions generating clusters and introduces a relevant matrix.

**Definition 4.7 (Cluster functions and matrix).** Let parameters $\mathfrak{M} \in \mathcal{L}$. The cluster functions associated to linear parameters $\mathfrak{M}$ are the functions $\mathfrak{h}_{m'}^m$ for $m, m' \in \mathcal{M}$ defined in Theorem 4.5 by, for $s < t$,

$$ \mathfrak{h}_{m'}^m(s, t) = \mathfrak{h}_{m'}^m(t-s) + \sum_{p, p' \in \mathcal{P}} \int_{s+}^{t-} \int_{s+}^{k} a_p^m \mathfrak{h}_p^m(t-k) \langle p, k \rangle_{\mathfrak{M}} \mathfrak{h}_{m'}^{p'}(k' - s) d\mathcal{D}_{k'}^{p'} d\mathcal{D}_k^p. $$

For $\mathfrak{M} \in \mathcal{L}$, the matrix $\mathfrak{H}_{\mathfrak{M}}$ is defined by

$$ (4.7) \qquad \mathfrak{H}_{\mathfrak{M}} = \left( \sup_{s \in \mathbb{R}} \| \mathfrak{h}_{m'}^m(s, \cdot) \|_{1, (s, \infty)} \right)_{m, m' \in \mathcal{M}}. $$

Let parameters $\mathfrak{M} \in \mathcal{L}$, we are interested in the clusters generated by the cluster functions $\mathfrak{h}_{m'}^m$ defined in Definition 4.7. As we will see later, the matrix $(\mathfrak{H}_{\mathfrak{M}})^T$, defined in Definition 4.7, gives a uniform bound on the expectation of the offspring distribution of clusters. This section follows [41].

Let

$$ \mathfrak{h} = (\mathfrak{h}_{m'}^m)_{m, m' \in \mathcal{M}}. $$

Clusters for linear HAR processes correspond to $\mathcal{P}$ois($\mathfrak{h}^T$) clusters in [41]. We explain below what it means, but for further details we refer the reader to [41].

**Remark 4.8.** The relevant parameter is $\mathfrak{h}^T$ and not $\mathfrak{h}$ since $\mathfrak{h}_{m'}^m$ corresponds to how many (in expectation) children of type $m$ a point from node $m'$ will have, whereas, in the proof of Theorem 3.9 for example, $\mathfrak{h}_{m'}^m$ corresponds to how much previous points of node $m'$ modify the intensity of node $m$. This small change of perspective, backward or forward, is the explanation on why $\mathfrak{h}^T$ and $(\mathfrak{H}_{\mathfrak{M}})^T$ are the relevant quantities for clusters.

We use the Ulam Harris Neveu notation to label the individuals of trees, see (2), and thus a tree is viewed as a subset of $\mathcal{U} = \cup_{n \in \mathbb{N}} (\mathbb{N}^*)^n$ with $(\mathbb{N}^*)^0 = \{\emptyset\}$.
Given parameters $\mathfrak{M} \in \mathcal{L}$, a cluster $G_{t_0}^{m_0}$ with root of type $m_0 \in \mathcal{M}$ and born at time $t_0 \in \mathbb{R}$ is defined by $((u, tp(u), bd(u)))_{u \in \mathcal{T}}$ where

* $\mathcal{T} \subset \mathcal{U}$ is a random tree. For $u \in \mathcal{T}$, $tp(u) \in \mathcal{M}$ is the type of $u$ and $bd(u) \in \mathbb{R}$ is the birth date of $u$.

* $\mathcal{T}$ has root $\emptyset$ of type $m_0$ (ie $tp(\emptyset) = m_0$) and born at time $t_0$ (ie $bd(\emptyset) = t_0$).

* Independently each individual $u \in \mathcal{T}$ reproduces as follows. If we denote $X(u, m)$ the number of children of $u$ with type $m \in \mathcal{M}$, then for $m \in \mathcal{M}$ the random variables $X(u, m)$ are independent and

$$X(u, m) \sim \mathcal{Poi}_s (\|\mathfrak{h}_{\text{tp}(u)}^m (\text{bd}(u), \text{bd}(u) + \cdot)\|_{1,(0,\infty)}) \quad \forall m \in \mathcal{M}.$$

The birth dates of the children of $u$ are distributed as follows. Conditionally to the children of $u$ and their types, independently for any child $v \in \mathcal{T}$ of $u$, we have

$$\text{bd}(v) \sim \frac{\mathfrak{h}_{\text{tp}(u)}^{\text{tp}(v)}(\text{bd}(u), t)}{\|\mathfrak{h}_{\text{tp}(u)}^{\text{tp}(v)}(\text{bd}(u), \text{bd}(u) + \cdot)\|_{1,(0,\infty)}} \mathbb{1}_{t>\text{bd}(u)}dt.$$

One may remark that $\mathfrak{h}^m \text{tp}(v)_{\text{tp}(u)} = (\mathfrak{h}^T)_{\text{tp}(u), \text{tp}(v)}$, thus, as in [41], row indexes correspond to father/mother and column indexes correspond to children.

Finally, if one omits the birth dates of a cluster $G$, it remains $(u, \text{tp}(u))_{u \in \mathcal{T}}$. It is clear that $(u, \text{tp}(u))_{u \in \mathcal{T}}$ can be stochastically dominated by $(u, \text{tp}_A(u))_{u \in \mathcal{A}}$ (which means that $\mathcal{T} \subset \mathcal{A}$ and for $u \in \mathcal{T}$, we have $\text{tp}(u) = \text{tp}_A(u)$) where $(u, \text{tp}_A(u))_{u \in \mathcal{A}}$ is a $\mathcal{Poi}_s(\mathfrak{h}_{\mathfrak{M}}^T)$ Galton Watson process. It means that independently, any point $u \in \mathcal{A}$ with type (in $\mathcal{A}$) $\text{tp}_A(u)$ reproduces as follows: independently for all $m \in \mathcal{M}$ it has $\mathcal{Poi}_s([\mathfrak{h}_{\mathfrak{M}}^T]_{\text{tp}_A(u),m})$ children with type $m$.

<figure><img src="image_4.png" /><figcaption>FIG 2. Illustration of a cluster. The tree $\mathcal{T}$ is represented with the types and birth dates. The ancestor is $u = \emptyset$, it has two children, $u = 0$ and $u = 1$. $u = 0$ has one child ($u = 00$), $u = 1$ has three children ($u = 10$, $u = 11$ and $u = 12$) and so on.</figcaption></figure>

Given this precise structure of clusters, the cluster representation takes the following form.

**Proposition 4.9 (Cluster representation).** Let $\mathfrak{M} \in \mathcal{L}$ some parameters. Suppose that we are under the same assumptions as Theorem 4.5. Let $(\Pi^m)_{m \in \mathcal{M}}$ be point processes such that, conditionally to $(\xi_k^p)_{(p,k) \in \mathcal{Q}}$, $(\Pi^m)_{m \in \mathcal{M}}$ are independent inhomogeneous Poisson processes with intensities given by the immigrant rates in Theorem 4.5. Conditionally to $(\Pi^m)_{m \in \mathcal{M}}$ let independent $\mathcal{Poi}_s(\mathfrak{h}^T)$ clusters $G_x^m$ with root of type $m$ and born at time $x$ for $m \in \mathcal{M}$ and $x \in \Pi^m$. Then we have the following equality in distribution

$$(S^m)_{m \in \mathcal{M}} \stackrel{d}{=} \{\{s \in \mathbb{R} \mid \exists m' \in \mathcal{M}, \exists x \in \Pi^{m'}, \exists u \in \mathcal{U} \text{ such that } (u, m, s) \in G_x^{m'}\}\}_{m \in \mathcal{M}}$$

where $(S^m)_{m \in \mathcal{M}}$ is the $\mathbf{HAR}_{\mathfrak{M}}(-\infty)$ process from Theorem 3.9.
Obtaining controls on clusters is crucial to derive properties for the whole process. For example, exponential bounds on clusters lead to exponential moments of number of points for the whole process, as explained in Section 5.

Deriving results on clusters requires assumptions on the cluster functions $h_m^m$. As for classical Galton Watson processes, the relevant assumption is about the spectral radius of the matrix that collects the size of the interactions. Let us introduce the assumption in our case.

**Assumption (Spec2).** We say that $\mathfrak{M} \in \mathcal{L}$ satisfies (Spec2) if the spectral radius of the matrix $\mathfrak{H}_{\mathfrak{M}}$ defined by (4.7) is smaller than one, equivalently $\text{SpR}(\mathfrak{H}_{\mathfrak{M}}) < 1$. We also require $\|h_m^p\|_{1,1}^\infty < \infty$ for all $m \in \mathcal{M}$ and all $p \in \mathcal{P}$.

**Remark 4.10.** Since the spectrum of a matrix and its transpose matrix is the same, in assumption (Spec2) it is equivalent to require $\text{SpR}(\mathfrak{H}_{\mathfrak{M}}^T) < 1$, which is the condition to have subcritical clusters in [41].

Under assumption (Spec2) one can already derive exponential controls on the number of points in a cluster. This result is a straightforward consequence of exponential moments for Galton Watson trees. An explicit bound, that depends on the matrix $\mathfrak{H}_{\mathfrak{M}}^T$ is derived in Theorems 3.1 and 4.1 of [41] under the condition $\text{SpR}(\mathfrak{H}_{\mathfrak{M}}^T) = \text{SpR}(\mathfrak{H}_{\mathfrak{M}}) < 1$.

# 5. Exponential moments of HAR processes

Obtaining good controls on the moments of a process is key point for both its theoretical study and applications. In this section we derive exponential moments for HAR processes. The main idea is to prove exponential moments them for HAR processes with linear parameters $\mathfrak{M} \in \mathcal{L}$ by taking advantage of cluster representation and then to transfer the result to general HAR processes by using a domination by a linear HAR process.

## 5.1. Linear domination of HAR processes

Given some parameters $\mathfrak{M}$ we want to dominate a HAR process with parameters $\mathfrak{M}$ by another HAR process with parameters $\mathfrak{M}^+ \in \mathcal{L}$. To do so, for any $\mathfrak{M}$ we introduce the associated dominating parameters $\mathfrak{M}^+ \in \mathcal{L}$. Of course, the random drifts will also have to be slightly modified since we are not assuming positiveness of the random drifts $\xi = (\xi_k^p)_{(p,k) \in \mathcal{Q}}$.

**Definition 5.1 (L domination).** Let some parameters $\mathfrak{M}$, possibly nonlinear, given by

$$ \mathfrak{M} = ((\mu_m)_m, (h_\beta^\alpha)_{\alpha,\beta}, (\Phi^\alpha)_\alpha, (\kappa_p^\alpha)_{\alpha,p}) $$

We define the parameters $\mathfrak{M}^+ \in \mathcal{L}$ and the new random drifts by the following tabular.

<table><thead><tr><th>Μ</th><th>μ<sub>m</sub></th><th>h<sub>β</sub><sup>α</sup></th><th>Φ<sup>α</sup></th><th>κ<sub>p</sub><sup>α</sup></th><th>ξ<sub>k</sub><sup>p</sup></th></tr></thead><tbody><tr><td>Μ<sup>+</sup></td><td>ν<sub>m</sub> := Φ<sup>m</sup>(μ<sub>m</sub>)</td><td>g<sub>β</sub><sup>α</sup> := L<sub>α</sub>|h<sub>β</sub><sup>α</sup>|</td><td>id</td><td>κ<sub>p</sub><sup>α</sup> := a<sub>p</sub><sup>α</sup> id + |κ<sub>p</sub><sup>α</sup>(0)|</td><td>ζ<sub>k</sub><sup>p</sup> := |ξ<sub>k</sub><sup>p</sup>|</td></tr></tbody></table>

We denote $b_p^\alpha := |\kappa_p^\alpha(0)|$ and by convention we still have $\kappa_p^\alpha(\emptyset) = 0$.

For any initial condition $\mathfrak{C}$ at time $t_0$, we define the initial condition $\mathfrak{C}^+$ at time $t_0$ by $\mathfrak{C}^{+,m} = \mathfrak{C}^m$, for all $m \in \mathcal{M}$ and for all $(p, k) \in \mathcal{Q}$ with $k < t_0$, $\mathfrak{C}_k^{+,p} = \emptyset$ if $\mathfrak{C}_k^p = \emptyset$ and $\mathfrak{C}_k^{+,p} = |\mathfrak{C}_k^p|$ if $\mathfrak{C}_k^p \in \mathbb{R}$.

The following proposition links HAR processes with parameters $\mathfrak{M}$ (and random drifts $\xi$) and the ones with parameters $\mathfrak{M}^+$ (and the random drifts $\zeta = |\xi|$). Recall that we denote by $\ell^m$ the pre-intensity defined by (2.3) so that $\lambda^m = \Phi^m(\ell^m)$.

**Proposition 5.2 (L domination).** Let some parameters $\mathfrak{M}$ and the associated $\mathfrak{M}^+$ defined in Definition 5.1.

1. Suppose that assumption (NoExpl)[$\mathfrak{M}$] holds, then (NoExpl)[$\mathfrak{M}^+$] also holds. For any $t_0 \in \mathbb{R}$, for any integrable initial condition $\mathfrak{C}$, the initial condition $\mathfrak{C}^+$ is also integrable and then the processes $\text{HAR}_{\mathfrak{M}}(t_0, \mathfrak{C}, \xi)$ and $\text{HAR}_{\mathfrak{M}^+}(t_0, \mathfrak{C}^+, \zeta)$ from Theorem 3.4 compare as follows:

$$ - \forall m, \forall t \ge t_0, \lambda_t^m \le (\lambda_t^m)_{\mathfrak{M}^+} \text{ and } L_m |\ell_t^m - \mu_m| \le (\lambda_t^m)_{\mathfrak{M}^+} - \Phi^m(\mu_m), $$

$$ - \forall m, \forall C \subset [t_0, \infty), S^m(C) \le (S^m(C))_{\mathfrak{M}^+}, $$

$$ - \forall (p, k) \in \mathcal{Q} \text{ with } k \ge t_0, |W_k^p| \le (W_k^p)_{\mathfrak{M}^+}. $$
2. Suppose that assumptions (Spec1) $[\mathfrak{M}]$ and (FiMe) $[\xi]$ hold, then (Spec1) $[\mathfrak{M}^+]$ and (FiMe) $[\zeta]$ also hold and then the processes $\text{HAR}_{\mathfrak{M}}(-\infty, \xi)$ and $\text{HAR}_{\mathfrak{M}^+}(-\infty, \zeta)$ from Theorem 3.9 compare as follows:

$$ - \forall m, \forall t \in \mathbb{R}, \lambda_t^m \le (\lambda_t^m)_{\mathfrak{M}^+} \text{ and } L_m |\ell_t^m - \mu_m| \le (\lambda_t^m)_{\mathfrak{M}^+} - \Phi^m(\mu_m), $$

$$ - \forall m, \forall C \subset \mathbb{R}, S^m(C) \le (S^m(C))_{\mathfrak{M}^+}, $$

$$ - \forall (p, k) \in \mathcal{Q}, |W_k^p| \le (W_k^p)_{\mathfrak{M}^+}. $$

This proposition allows us to transfer results on linear HAR processes to general HAR processes.

## 5.2. Exponential moments

In this subsection we use results on linear HAR processes, cluster representation and results on clusters, to derive exponential moments for linear HAR processes. Then we transfer these results to general HAR processes using the previous subsection. Having exponential moments cannot be achieved without an assumption on the random drifts. We will suppose the following sub-gaussian assumption.

**Assumption (SubG).** There exist constants $\epsilon_p, \delta_p > 0$ for $p \in \mathcal{P}$ such that for all $(p, k) \in \mathcal{Q}$, for all $t \ge 0$ we have,

$$ \mathbb{E}[e^{t|\xi_k^p|}] \le e^{\epsilon_p t + \delta_p t^2}. $$

Let $F = (F^\alpha)_{\alpha \in \mathcal{M} \cup \mathcal{P}}$ be non-negative functions. For $X = (\lambda, S, W)$ a HAR process we define $F \bullet S$ and $F \bullet |W|$ by,

$$ F \bullet S = \sum_{m \in \mathcal{M}} \int_{\mathbb{R}} F^m(t) dS_t^m, \quad F \bullet |W| = \sum_{p \in \mathcal{P}} \int_{\mathbb{R}} F^p(k) |W_k^p| d\mathcal{D}_k^p. $$

Of course, one may omit the absolute values on $W$ in case of a linear HAR process. Let $n \in \mathbb{N}^*$. We also define the following vectors,

$$ F_{\infty, n}^{\mathcal{M}} = (\|F^m\|_{1,1/n}^{\infty})_{m \in \mathcal{M}}, \quad F_{\infty}^{\mathcal{P}} = (\|F^p\|_{1,n_p}^{\infty})_{p \in \mathcal{P}}. $$

We use the same notations as in [41], so that for $\mathfrak{M} \in \mathcal{L}$, the log Laplace transform of a $\text{Pois}(\mathfrak{H}_{\mathfrak{M}}^\mathcal{T})$ Galton Watson tree is the function $\mathcal{L}: \mathbb{R}_+^M \to (\mathbb{R}_+ \cup \{\infty\})^M$, defined by

$$ \mathcal{L}(u)_m = \log \left( \mathbb{E} \left[ e^{u \cdot \text{Card}_M(\mathcal{T}^m)} \right] \right) $$

where $u \in \mathbb{R}_+^M, \mathcal{T}^m$ is a $\text{Pois}(\mathfrak{H}_{\mathfrak{M}}^\mathcal{T})$ Galton Watson tree with root of type $m \in \mathcal{M}$ and $\text{Card}_M(\mathcal{T}^m) \in \mathbb{N}^M$ is the vector containing the number of individuals of each type in $\mathcal{T}^m$.

**Theorem 5.3** (Exponential moments for linear HAR processes). Let $\mathfrak{M} \in \mathcal{L}$ and $(\lambda, S, W)$ the HAR process $\text{HAR}_{\mathfrak{M}}(-\infty)$ from Theorem 3.9. Suppose that

* The assumptions of Theorem 3.9 hold: (Spec1) $[\mathfrak{M}]$ and (FiMe).

* Assumptions (Spec2) $[\mathfrak{M}]$ and (SubG) hold (which implies assumption (FiMe)).

For any $n \in \mathbb{N}^*$, let $R_n = (\|h_m^p\|_{1,1/n}^\infty)_{(m,p) \in \mathcal{M} \times \mathcal{P}} \times [(\text{Id} - H_W^W)^{-1}]^\mathcal{T} \prec \infty$ and define

$$ \chi_n^M = n(e^{\mathcal{L}(F_{\infty,n}^M)} - 1), $$

$$ \chi_n^P = n(e^{\mathcal{L}(R_n F_{\infty}^P)} - 1), $$

where $\mathcal{L}$ is the log Laplace function of a $\text{Pois}(\mathfrak{H}_{\mathfrak{M}}^\mathcal{T})$ Galton Watson process. Then there exist $0 < K < \infty$ and a matrix $A_\infty$ with non negative entries, both depending only on $\mathfrak{M}$ such that the following holds for all $n \in \mathbb{N}^*$,

$$ \mathbb{E}[e^{F \bullet S}] \le \exp \left[ K \left( |\chi_n^M|_1 + |A_\infty \chi_n^M|_1 + |A_\infty \chi_n^M|_2^2 \right) \right], $$

$$ \mathbb{E}[e^{F \bullet W}] \le \exp \left[ K \left( |F_{\infty}^P|_1 + |\chi_n^P|_1 + |\tilde{F}_{\infty}^P + A_{\infty} \chi_n^P|_1 + |\tilde{F}_{\infty}^P + A_{\infty} \chi_n^P|_2^2 \right) \right], $$
with $\tilde{F}_{\infty}^{P} := [(\mathrm{Id} - H_{W}^{W})^{-1}]^{\mathrm{T}} F_{\infty}^{P}$. Moreover, one can take

$$A_{\infty} = [(\mathrm{Id} - H_{W}^{W})^{-1}]^{\mathrm{T}} (a_{p}^{m} \| h_{p}^{m} \|_{1, n_{p}}^{\infty})_{p, m \in \mathcal{P}, \mathcal{M}} \prec \infty.$$

The assumptions are discussed after Corollary 5.5, stating exponential moments for general parameters.

**Remark 5.4.** The function $\mathcal{L}$ is deeply studied in [41]. Without diving into too much details, under assumption (Spec2), there exist $0 < \theta_0, \theta_1 < \infty$, depending on $\mathfrak{M}$, such that if $|u|_{\infty} \le \theta_0$ then $|\mathcal{L}(u)|_{\infty} \le \theta_1$ [41]. Thus for any $F$, as long as $F_{\infty,n}^{M}, F_{\infty}^{P} < \infty$, one can find $\theta > 0$ such that $e^{\theta F \cdot S}$ and $e^{\theta F \cdot W}$ have finite expectation.

The $\ell^2$ norms come from the term $\|\zeta_p t^2\|$ in assumption (SubG). The matrix $A_{\infty}$ correspond to the contribution of the random drifts to the immigrant rates in the cluster representation. The quantities $F_{\infty,n}^{M}$ (resp. $R_n F_{\infty}^{P}$) summarize the total weights allocated to points in $F \cdot S$ (resp. $F \cdot W$) in the cluster representation. Then the key quantities to calculate the exponential moments are respectively $\mathcal{L}(F_{\infty,n}^{M})$ and $\mathcal{L}(R_n F_{\infty}^{P})$.

The entries of the matrix $R_n$ and the vector $F_{\infty,n}^{M}$ are decreasing with $n \in \mathbb{N}^*$ so one can find the best trade-off in $n$ between this decrease and the factor $n$ in front of $\chi_n^M$ and $\chi_n^P$.

From Theorem 4.5 it is clear that for a model $\mathfrak{M} \in \mathcal{L}$, the stationary process $\mathbf{HAR}_{\mathfrak{M}}(-\infty)$ dominates all processes $\mathbf{HAR}_{\mathfrak{M}}(t_0)$ with $t_0 \in \mathbb{R}$. Thus this result is also valid for linear HAR processes started at time $t_0 \in \mathbb{R}$ with empty initial condition. A similar result can be stated for HAR processes started at time $t_0$ with an initial condition, up to moments assumptions on the initial condition.

The following result states exponential moments for general HAR processes. It is a direct consequence of the linear domination stated in Proposition 5.2 and the exponential moments for linear HAR processes stated in Theorem 5.3.

**Corollary 5.5 (Exponential moments).** Let $\mathfrak{M}$ general parameters and $(\lambda, S, W)$ the HAR process $\mathbf{HAR}_{\mathfrak{M}}(-\infty)$ from Theorem 3.9. Suppose that

* Assumptions (Spec1)[$\mathfrak{M}$] and (Spec2)[$\mathfrak{M}^+$] hold ($\mathfrak{M}^+$ from Definition 5.1).

* Assumption (SubG) holds.

For any $n \in \mathbb{N}^*$, define the matrix

$$R_n = (L^p \|h_m^p\|_{1,1/n}^{\infty})_{(m,p) \in \mathcal{M} \times \mathcal{P}} \times [(\mathrm{Id} - H_W^W)^{-1}]^{\mathrm{T}}.$$

Then, there exists $\theta_0 > 0$ and $K < \infty$ depending on $\mathfrak{M}$ such that,

$$\forall n \in \mathbb{N}^*, \quad \frac{|F_{\infty,n}^M|_{\infty}}{\theta_0} := \beta \le 1 \quad \implies \quad \mathbb{E}[e^{F \cdot S}] \le e^{K(\beta n + \beta^2 n^2)},$$

and

$$\forall n \in \mathbb{N}^*, \quad \frac{|R_n F_{\infty}^P|_{\infty}}{\theta_0} := \beta \le 1 \quad \implies \quad \mathbb{E}[e^{F \cdot |W|}] \le e^{K[\beta n + \beta^2 n^2 + |F_{\infty}^P|_{\infty} + |F_{\infty}^P|^2_{\infty}]}.$$

Some remarks are in order.

Let $L \in \mathbb{N}^*$. If one takes $F^m = \theta_0 1_{[0,L)}$ for all $m \in \mathcal{M}$, and choose $n = L$ then it follows that

$$F \cdot S = \theta_0 \sum_{m \in \mathcal{M}} S^m([0, L)) =: \theta_0 S([0, L)),$$

and $F_{\infty,n}^{M} = \theta_0 1$. Thus we obtain,

$$\mathbb{E}[e^{\theta_0 S([0, L])}] \le e^{KL^2} < \infty,$$

with $K$ a constant depending on $\mathfrak{M}$. Similarly, there exists $\theta_2 > 0$ depending on $\mathfrak{M}$, such that by taking $F^p(k') = \theta_2 1_{k'=k}$ for $k \in \mathcal{D}_p$, we obtain,

$$\sup_{(p,k) \in \mathcal{Q}} \mathbb{E}[e^{\theta_2 |W_k^p|}] < \infty.$$

The $n^2$ (or $L^2$) dependency is closely linked to the assumption (SubG).
If $P = \emptyset$, a HAR process is a classical Hawkes process and the assumptions are now reduced to $\text{SpR}(H_S^S) < 1$ (since $H_{\mathfrak{M}} = \mathfrak{H}_{\mathfrak{M}+} = H_S^S$ in this case). Hence we recover the assumptions of [40] and [41] about exponential moments for Hawkes processes.

Now we focus on the main question, which is whether or not (Spec1)[M] (assumption to have existence of stationary processes) implies (Spec2)[M+] (assumption to have exponential moments of clusters). It is the case for Hawkes processes since in this case $H_{\mathfrak{M}} = \mathfrak{H}_{\mathfrak{M}+} = H_S^S$. For HAR processes it is more complex. Let parameters M such that (Spec1)[M] holds, ie $\text{SpR}(H_{\mathfrak{M}}) < 1$ with $H_{\mathfrak{M}}$ defined in Definition 3.8 by

$$H_{\mathfrak{M}} = \begin{pmatrix} H_S^S & H_W^S \\ H_S^W & H_W^W \end{pmatrix}.$$

Then (Spec1)[M+] also holds with M+ by Proposition 5.2. Assumption (Spec2)[M+] is $\text{SpR}(\mathfrak{H}_{\mathfrak{M}+}) < 1$ with $\mathfrak{H}_{\mathfrak{M}+}$ defined by (4.7). The bound (4.5) given in Theorem 4.5 implies that

$$(5.1) \qquad \mathfrak{H}_{\mathfrak{M}+} \preceq H_S^S + \tilde{H}_W^S (\text{Id} - H_W^W)^{-1} \tilde{H}_S^W$$

where $\tilde{H}_W^S = (n_p L_m a_p^m \|h_p^m\|_1)_{m,p}$ and $\tilde{H}_S^W = (L_p \|h_m^p\|_{1,1}^\infty)_{p,m}$ and $\preceq$ means the entry by entry comparison. The difference between $(\tilde{H}_W^S, \tilde{H}_S^W)$ and $(H_W^S, H_S^W)$ is that when the classic $L^1$ norm is used on one side, the discrete $L_1^\infty$ norm is used on the other side. Indeed,

$$H_W^S = (L_m a_p^m \|h_p^m\|_{1,n_p}^\infty)_{m,p} \quad \text{and} \quad H_S^W = (L_p \|h_m^p\|_1)_{p,m}.$$

This is due to the fact that we are integrating in the past (between $-\infty$ and $t$) to establish the existence of the stationary HAR processes (leading to $H_{\mathfrak{M}}$) while we are integrating in the future (between $s$ and $\infty$) to describe clusters (leading to $\mathfrak{H}_{\mathfrak{M}+}$).

On the one hand, $\tilde{H}_W^S$ and $H_W^S$ compare in the right sense: $\tilde{H}_W^S \preceq H_W^S$ since $n_p \|\cdot\|_1 \le \|\cdot\|_{1,n_p}^\infty$ but on the other hand, we have $\tilde{H}_S^W \succeq H_S^W$, the wrong sense, since $\|\cdot\|_1 \le \|\cdot\|_{1,1}^\infty$. Thus we cannot conclude on (Spec1)[M] $\Rightarrow$ (Spec2)[M+]. However, we are able to prove this comforting statement.

**Proposition 5.6.** Let parameters M. Then we have

$$\text{SpR} \left( \tilde{H}_{\mathfrak{M}} := \begin{pmatrix} H_S^S & H_W^S \\ \tilde{H}_S^W & H_W^W \end{pmatrix} \right) < 1 \implies \text{(Spec1)[M]} \text{ and } \text{(Spec2)[M+]}.$$

**Remark 5.7.** For $M \in \mathcal{L}$, the condition $\text{SpR}(\tilde{H}_{\mathfrak{M}}) < 1$ ensures both existence of the stationary process and subcritical clusters.

# 6. Asymptotic limits and application to concentration inequalities

In the study of stochastic processes, understanding whether or not a process converges to a stationary one plays a key role. A positive answer allows to transfer questions about the non stationary process into the study of the stationary one. It also opens the door to some creative ideas such as the coupling presented at the end of this section. This coupling ultimately leads to powerful concentration inequalities with numerous applications, such as in statistics.

## 6.1. Rate of convergence to the stationary state

Our main interest is about the convergence of HAR processes started at some time $t_0 \in \mathbb{R}$ to the stationary HAR process. To remain as general as possible, the main theorem of this section states the convergence of two HAR processes with distinct initial conditions towards each other. Then one can specify this result with the empty initial condition and the stationary initial condition to obtain the convergence of the non stationary process to the stationary one. This is done in Corollary 6.4.

Let us introduce the following assumption.
**Assumption (ExpTail).** We say that $\mathfrak{M}$ satisfies assumption (ExpTail) if there exist constants $K, c > 0$ such that for all $\alpha, \beta \in \mathcal{M} \cup \mathcal{P}$, for all $x > 0$,

$$|h_{\beta}^{\alpha}(x)| \leq K e^{-cx}.$$

This tail assumption is the starting point to derive rate of convergence of two HAR processes with different initial conditions towards each other. This tail assumption transfers to coefficients and clusters functions introduced in Section 4 as explained in the following result.

**Proposition 6.1.** Let parameters $\mathfrak{M} \in \mathcal{L}$, suppose that (Spec1.1) $[\mathfrak{M}]$ and (ExpTail) $[\mathfrak{M}]$ hold. Then there exist constants $K, c > 0$ such that

$$\left\{ \begin{array}{c} p, k \\ p', k' \end{array} \right\}_{\mathfrak{M}} \leq K e^{-c(k-k')}, \quad \forall (p, k), (p', k') \in \mathcal{Q} \text{ with } k' \leq k$$

and

$$h_{m'}^{m}(s, t) \leq K e^{-c(t-s)}, \quad \forall m, m' \in \mathcal{M} \text{ and } s < t.$$

The corner stone of this section is having precise control on the tail of clusters. Assumption (ExpTail) on the tails of the interaction functions leads to the following lemma, which is a specific case of Theorem 4.1 from [41]. As for point processes, for any cluster $G_s^m$ born at time $s$ with a root of type $m$, for any $I \subset \mathbb{R}$, we denote by $G_s^m(I)$ the total number of individuals of $G_s^m$ whose birth dates are in $I$.

**Lemma 6.2.** Let parameters $\mathfrak{M} \in \mathcal{L}$ and suppose that assumptions (ExpTail) $[\mathfrak{M}]$ and (Spec2) $[\mathfrak{M}]$ hold. Let $G_{t_0}^m$ be a $\mathcal{P}$ois($h^T$) cluster with root of type $m \in \mathcal{M}$ and born at time $t_0 \in \mathbb{R}$. Then, there exists $\theta_0 > 0$ depending on $\mathfrak{M}$ such that for all $t \geq 0$,

$$E \left[ e^{\theta_0 G_{t_0}^m ([t_0+t, \infty))} \right] \leq 1 + K \exp(-ct)$$

for constants $c > 0$, $K > 0$ depending on $\mathfrak{M}$ but not on $m$ nor $t_0$.

Given two initial conditions $\mathfrak{C}_1$ and $\mathfrak{C}_2$ at time $t_0$ and $\delta > 0$ we define $|\mathfrak{C}_1, \mathfrak{C}_2|_{\delta}$ as follows,

$$|\mathfrak{C}_1, \mathfrak{C}_2|_{\delta} := \max \left( \left[ \max_{m \in \mathcal{M}} \int_{-\infty}^{t_0^{-}} e^{-\delta(t_0-s)} d(\mathfrak{C}_1^m \Delta \mathfrak{C}_2^m)_s \right], \left[ \max_{p \in \mathcal{P}} \int_{-\infty}^{t_0^{-}} e^{-\delta(t_0-k)} \psi(\mathfrak{C}_{1,k}^p, \mathfrak{C}_{2,k}^p) d\mathcal{D}_k^p \right] \right)$$

with $\Delta$ the symmetric difference on sets and $\psi$ defined by

$$(6.1) \qquad \psi(x, y) := \begin{cases} |x - y| & \text{if } x, y \in \mathbb{R} \\ 0 & \text{if } x = y = \emptyset \\ |x| + 1 & \text{if } x \in \mathbb{R}, y = \emptyset \\ 1 + |y| & \text{if } x = \emptyset, y \in \mathbb{R} \end{cases}.$$

The idea is to measure how different the two initial conditions are. Since the $\mathfrak{C}_k^p$ are allowed to take both real values and the value $\emptyset$, we need the function $\psi$ to deal with this two types of possible values. The following theorem is the main result of this section.

**Theorem 6.3 (Asymptotic limit).** Let parameters $\mathfrak{M}$ and suppose that assumptions (NoExpl) $[\mathfrak{M}]$, (Spec1.1) $[\mathfrak{M}]$, (ExpTail) $[\mathfrak{M}]$ and (Spec2) $[\mathfrak{M}^+]$ hold. Let $t_0 \in \mathbb{R}$ and two integrable initial conditions, $\mathfrak{C}_1$ and $\mathfrak{C}_2$, at time $t_0$ for $\mathfrak{M}$. For $i = 1, 2$, we denote by

$$((\lambda^m, \mathfrak{C}_i)_{m \in \mathcal{M}}, (S^m, \mathfrak{C}_i)_{m \in \mathcal{M}}, (W^p, \mathfrak{C}_i)_{p \in \mathcal{P}})$$

the unique non exploding HAR$_{\mathfrak{M}}(t_0, \mathfrak{C}_i)$ process from Theorem 3.4. There exists $\delta > 0$ depending only on $\mathfrak{M}$ such that the following holds.

* There exist $K, c > 0$ depending only on $\mathfrak{M}$ such that for all $t \geq 0$,

$$P\left(\forall m \in \mathcal{M}, (S^m, \mathfrak{C}_1 \Delta S^m, \mathfrak{C}_2) \cap [t_0 + t, \infty) = \emptyset\right) \geq 1 - KE\left[|\mathfrak{C}_1, \mathfrak{C}_2|_{\delta}\right] e^{-ct},$$
$$ \mathbb{P}\left(\sum_{p \in \mathcal{P}} \int_{t_0+t}^{\infty} |W_k^{p, \mathcal{C}_1} - W_k^{p, \mathcal{C}_2}| d\mathcal{D}_k^p \le e^{-ct}\right) \ge 1 - K \mathbb{E}[|\mathcal{C}_1, \mathcal{C}_2|_{\delta}] e^{-ct}. $$

* The following convergences hold a.s. and in $L^q$ for $q \ge 1$ such that there exists $q_0 > q$ satisfying $|\mathcal{C}_1, \mathcal{C}_2|_{\delta} \in L^{q_0}$;

$$ \sum_{m \in \mathcal{M}} \mathrm{Card}((S^{m, \mathcal{C}_1} \Delta S^{m, \mathcal{C}_2}) \cap [t+t_0, \infty)) \xrightarrow{t \to \infty} 0, $$

$$ \sum_{p \in \mathcal{P}} \int_{t_0+t}^{\infty} |W_k^{p, \mathcal{C}_1} - W_k^{p, \mathcal{C}_2}| d\mathcal{D}_k^p \xrightarrow{t \to \infty} 0. $$

Moreover, the rates of $L^q$ convergence are at least $Ke^{-ct}$ with $K, c > 0$ constants depending on $\mathfrak{M}, q, q_0$ and $\mathbb{E}[|\mathcal{C}_1, \mathcal{C}_2|_{\delta}^{q_0}]$ for $K$.

We obtained convergence, almost surely, in $L^q$ and in probability, of $\mathbf{HAR}_{\mathfrak{M}}(t_0, \mathcal{C}_i)$, $i = 1, 2$ toward each other when time goes to infinity with explicit convergence rates. This shows that the asymptotic behavior does not depend on the initial condition and thus the process is decorrelated at infinity.

One can specify the result by taking for the first initial condition the empty initial condition and for the second one, the initial condition which is equal to the stationary HAR process. This allows to recreate the stationary HAR process through a HAR process with this particular initial condition. These two clever choices lead to the following result.

**Corollary 6.4** (Convergence to the 1-stationary HAR process). Let parameters $\mathfrak{M}$ and suppose that assumptions (Spec1)[$\mathfrak{M}$], (FiMe)[$\mathfrak{M}$], (ExpTail)[$\mathfrak{M}$] and (Spec2)[$\mathfrak{M}^+$] hold. Let $t_0 \in \mathbb{R}$. We denote by

$$ ((\lambda^{m,t_0})_{m \in \mathcal{M}}, (S^{m,t_0})_{m \in \mathcal{M}}, (W^{p,t_0})_{p \in \mathcal{P}}) $$

the $\mathbf{HAR}_{\mathfrak{M}}(t_0)$ process from Theorem 3.4 and

$$ ((\lambda^{m,-\infty})_{m \in \mathcal{M}}, (S^{m,-\infty})_{m \in \mathcal{M}}, (W^{p,-\infty})_{p \in \mathcal{P}}) $$

the $\mathbf{HAR}_{\mathfrak{M}}(-\infty)$ process from Theorem 3.9. Then the following holds.

* There exist $K, c > 0$ depending on $\mathfrak{M}$ such that for all $t \ge 0$,

$$ \mathbb{P}(\forall m \in \mathcal{M}, (S^{m,t_0} \Delta S^{m,-\infty}) \cap [t+t_0, \infty)) = \emptyset) \ge 1 - Ke^{-ct}, $$

$$ \mathbb{P}\left(\sum_{p \in \mathcal{P}} \int_{t_0+t}^{\infty} |W_k^{p,t_0} - W_k^{p,-\infty}| d\mathcal{D}_k^p \le e^{-ct}\right) \ge 1 - Ke^{-ct}. $$

* The following convergences hold a.s. and, under assumption (SubG), in $L^q$ for all $q \ge 1$

$$ \sum_{m \in \mathcal{M}} \mathrm{Card}((S^{m,t_0} \Delta S^{m,-\infty}) \cap [t+t_0, \infty)) \xrightarrow{t \to \infty} 0, $$

$$ \sum_{p \in \mathcal{P}} \int_{t_0+t}^{\infty} |W_k^{p,t_0} - W_k^{p,-\infty}| d\mathcal{D}_k^p \xrightarrow{t \to \infty} 0. $$

Moreover, the speeds of $L^q$ convergence are at least $Ke^{-ct}$ with $K, c > 0$ constants depending on $\mathfrak{M}$ and $q$.

This result links two of the most natural HAR processes: the stationary process and the process started at a fixed time $t_0$. They converge towards each other with exponential rates.
## 6.2. Berbee coupling, application to exponential concentration inequalities

For now, what remains unknown about HAR processes is whether or not they concentrate. More precisely, if functionals of the stationary HAR process concentrate around their means. For example, we can wonder if the total number of points in $[0, T)$ divided by $T \in \mathbb{N}^*$ tends to the expectation of the number of points in $[0, 1)$ as $T$ tends to $\infty$. More rigorously, do we have

$$ \frac{S([0, T])}{T} = \frac{1}{T} \sum_{k=1}^{T} S([k-1, k)) \rightarrow \mathbb{E}[S([0, 1))]? $$

By 1-stationary, for all $k \in \mathbb{N}$ we have $S([k-1, k)) \sim S([0, 1))$. Thus we only miss independence between the random variables $U_k = S([k-1, k))$ to apply concentration inequalities. However, by Theorem 6.3, we have decorrelation between $U_k$ and $U_{k+h}$ as $h \to \infty$. Indeed, Theorem 6.3 proves that memory fades exponentially fast. To make use of this "weak independence" structure we construct coupling a la Berbee [6]. Similar techniques were used for stationary time series, see [23, 51, 58].

**Definition 6.5** (Berbee coupling). Let parameters $\mathfrak{M}$. Suppose that assumption (NoExpl) $[\mathfrak{M}]$ holds. Let $n \in \mathbb{N}^*$, $0 < \ell \le n$ and $0 \le \phi < n$. Define the process

$$ \mathcal{X}^{(n, \ell, \phi)} = \left( (\mathfrak{X}^{m, (n, \ell, \phi)})_{m \in \mathcal{M}}, (\mathfrak{S}^{m, (n, \ell, \phi)})_{m \in \mathcal{M}}, (\mathfrak{W}^{p, (n, \ell, \phi)})_{p \in \mathcal{P}} \right) $$

by, $\forall j \in \mathbb{Z}$, on $[jn + \phi, jn + n + \phi)$ we have

$$ \mathcal{X}^{(n, \ell, \phi)} = X^{jn+\phi-\ell} $$

where the process $X^{jn+\phi-\ell}$ is the unique non exploding solution of $\mathbf{HAR}_{\mathfrak{M}}(jn + \phi - \ell)$.

The idea of this construction is to create a process close to the stationary HAR process and with more explicit independence properties. To do so we start a process at time $jn + \phi - \ell$, we let it evolve during a time $\ell$ and then we use it on $[jn + \phi, jn + n + \phi)$, and this for every $n \in \mathbb{Z}$.

<figure><img src="image_5.png" /><figcaption>FIG 3. Schematic representation of the coupling $\mathcal{X}^{(n,\ell,\phi)}$. The dotted part in red or green represent the unused part of the $\mathbf{HAR}_{\mathfrak{M}}(jn + \phi - \ell)$ processes. Only the solid lines are part of the coupling. The blue lines represent the independent random variables associated to $j-2$ and $j$ of Proposition 6.6.</figcaption></figure>

This coupling can be broken into independent blocks described in Proposition 6.6 below and illustrated in Figure 3.

**Proposition 6.6.** Let $n \in \mathbb{N}^*$, $0 < \ell \le n$, $0 \le \phi < n$ and $\mathcal{X}^{(n,\ell,\phi)}$ the coupling from Definition 6.5. For any integer $j$ we have,

$$ \mathcal{X}^{(n, \ell, \phi)}|_{[jn+\phi, \infty)} \perp \mathcal{F}_{(jn+\phi-\ell)-} $$

where $\mathcal{F}_{s-} = \sigma(\bigcup_{r<s} \mathcal{F}_r)$. Moreover, the random variables

$$ \mathcal{X}^{(n, \ell, \phi)}|_{[jn+\phi, (j+1)n+\phi+(n-\ell))} $$

for $j \in 2\mathbb{Z}$ are iid (independent and identically distributed). The same is true for $j \in 2\mathbb{Z} + 1$.
Proposition 6.6 is straightforward from the definition of the coupling. The second point shows how useful this coupling is in order to obtain concentration inequalities. Our coupling can be segmented into two interlaced sequences of iid random variables. However the two sequences are not independent between each others. The following proposition quantifies how close the stationary HAR process is from the coupling.

**Proposition 6.7.** Suppose that $\mathfrak{M}$ satisfies assumptions (Spec1) $[\mathfrak{M}]$, (FiMe), (ExpTail) $[\mathfrak{M}]$ and (Spec2) $[\mathfrak{M}^+]$. Let $n \in \mathbb{N}^*$, $0 < \ell \le n$, $0 \le \phi < n$ and $\mathcal{X}^{(n,\ell,\phi)}$ the coupling from Definition 6.5. Let also $X^{-\infty}$ the stationary $\mathbf{HAR}_{\mathfrak{M}}(-\infty)$ process. Then there exist constants $K, c > 0$ depending on $\mathfrak{M}$ only such that, for all integer $j$ we have

$$ \mathbb{P}\left(\forall m \in \mathcal{M}, \left(S^{m,(n,\ell,\phi)} \triangleq S^{m,-\infty}\right) \cap [jn + \phi, (j+1)n + \phi) = \emptyset\right) \ge 1 - K e^{-c\ell} $$

and

$$ \mathbb{P}\left(\sum_{p \in \mathcal{P}} \int_{jn+\phi}^{((j+1)n+\phi)-} |\mathcal{W}_k^{p,(n,\ell,\phi)} - \mathcal{W}_k^{p,-\infty}| d\mathcal{D}_k^p \le e^{-c\ell}\right) \ge 1 - K e^{-c\ell}. $$

This result is straightforward from Corollary 6.4 since we are comparing a non stationary process that has evolved a time $\ell$ to the stationary process. The larger $\ell$ is, the better the approximation of the stationary process by the coupling is, but the fewer independent pieces there are. Thus to prove concentration results for the stationary process there is a trade-off between how many independent pieces there are and how close the independent pieces are from the stationary process.

## 7. LASSO estimation for HAR processes

The collection of parameters $\mathfrak{M}$ of a HAR process contains relevant information about the underlying real phenomena being modeled. The aim of this section is to construct a LASSO estimator of the interaction functions and the base rates. We use a LASSO procedure to select variables and recover only non zero interactions. This section is an adaptation of [28] and [54].

### 7.1. Setting

To estimate the interactions with the LASSO procedure, we mainly follow the approach in [28], with some differences due to the autoregressive part. In this sequel, we adopt the following setting.

* We consider the case where the link functions $\Phi^\alpha$ for $\alpha \in \mathcal{M} \cup \mathcal{P}$ are known and are equal to the identity function.

* The masks are also assumed to be known, without any other requirement except that the intensity must be non negative. Assuming the masks to be known makes sense for applications. Indeed, the choices of the masks are inherent to the experiment being conducted: what type of interactions do we want to detect? For example, one can take $\kappa(W) = |W|$ if only the absolute value is relevant or $\kappa(W) = W \mathbb{1}_{W>0}$ if only positive signed $W$ should interact.

* The interaction functions are denoted $h_{\beta}^{\alpha,\star}$ for $\alpha, \beta \in \mathcal{M} \cup \mathcal{P}$. We put an exponent $\star$ to indicate that this parameter is unknown and has to be estimated. We are also assuming that the supports of the interaction functions are included in $(0, A]$ with $A > 0$ known, and that they belong to $L^2((0, A])$.

* The point processes base rates, $\mu_m^*$ for $m \in \mathcal{M}$ are also among the parameters to be estimated.

* The random drifts contains also relevant information since we do not assume them to be centered. From a statistical point of view their means are also relevant parameters. Thus we write

$$ \xi_k^p = \mu_p^* + \varepsilon_k^p, \quad (p, k) \in \mathcal{Q}, $$

with $\mu_p^* = \mathbb{E}[\xi_k^p]$. We also denote $\sigma_p^2 = \text{Var}(\xi_k^p) = \text{Var}(\varepsilon_k^p)$ for $(p, k) \in \mathcal{Q}$.

We summarize the above consideration in $\mathfrak{M}^*$ containing all the true parameters,

$$ (7.1) \qquad \mathfrak{M}^* = (\mu_{\alpha}^*, h_{\beta}^{\alpha,\star}, \Phi^{\alpha} = \text{id}, \kappa_{p}^{\alpha}, \sigma_{p}, A)_{\alpha, \beta \in \mathcal{M} \cup \mathcal{P}, p \in \mathcal{P}}, $$

and we suppose that (Spec1) $[\mathfrak{M}^*]$ holds. (FiMe) also holds since we assume the existence of $\mu_p^*$ for $p \in \mathcal{P}$. Thus if $X = (\lambda, S, W)$ is the $\mathbf{HAR}_{\mathfrak{M}^*}(-\infty)$ process from Theorem 3.9, it satisfies the following equations:
$$ (7.2a) \qquad \lambda_t^m = \mu_m^* + \sum_{m' \in \mathcal{M}} \int_{t-A}^{t-} h_{m'}^{m,*}(t-s) dS_s^{m'} + \sum_{p \in \mathcal{P}} \int_{t-A}^{t-} h_p^{m,*}(t-k) \kappa_p^m (W_k^p) d\mathcal{D}_k^p, $$

$$ (7.2b) \qquad S^m(C) = \int_{C \times \mathbb{R}_+} \mathbb{1}_{x \le \lambda_s^m} d\pi^m(s, x), $$

$$ (7.2c) \qquad W_k^p = \varepsilon_k^p + \mu_p^* + \sum_{m \in \mathcal{M}} \int_{k-A}^{k-} h_m^{p,*}(k-s) dS_s^m + \sum_{p' \in \mathcal{P}} \int_{k-A}^{k-} h_{p'}^{p,*}(k-k') \kappa_{p'}^p (W_{k'}^{p'}) d\mathcal{D}_{k'}^{p'}. $$

The following tabular summaries the known and unknown parameters.

<table><thead><tr><th>known/unknown</th><th>known</th><th>unknown</th><th>unknown</th><th>known</th></tr></thead><tbody><tr><th>parameter</th><td>Φ<sup>α</sup> for α ∈ M ∪ P</td><td>μ<sub>m</sub><sup>*</sup> for m ∈ M</td><td>h<sub>β</sub><sup>α,*</sup> for α, β ∈ M ∪ P</td><td>A</td></tr><tr><th>value/conditions</th><td>id</td><td>μ<sub>m</sub><sup>*</sup> ≥ 0</td><td>h<sub>m'</sub><sup>m,*</sup> ≥ 0 and h<sub>β</sub><sup>α</sup> ∈ L<sup>2</sup>((0, A])</td><td>A &lt; ∞</td></tr></tbody></table>

<table><thead><tr><th>known/unknown</th><th>known</th><th>unknown</th><th>possibly unknown</th></tr></thead><tbody><tr><th>parameter</th><td>κ<sub>p</sub><sup>α</sup> for α ∈ M ∪ P and p ∈ P</td><td>μ<sub>p</sub><sup>*</sup> = E[ξ<sub>k</sub><sup>p</sup>] for p ∈ P</td><td>σ<sub>p</sub><sup>2</sup> = Var(ξ<sub>k</sub><sup>p</sup>) for p ∈ P</td></tr><tr><th>value/conditions</th><td>κ<sub>p</sub><sup>m</sup> ≥ 0</td><td></td><td></td></tr></tbody></table>

We aim to estimate all unknown parameters denoted with an exponent ⋆ from the observation over a time window of the process X, more precisely an observation (S, W).

## 7.2. Oracle inequality

Let $T \ge 2$ an integer and suppose that we are observing the process X on the time window $[-A, T)$, which means that we observe $X_{-A,T} = [(S^m \cap [-A, T))}_{m \in M}$, $(W_k^p, k \in D_p \cap [-A, T))}_{p \in P}$.

Let $\mathfrak{N}^*$ the true parameters to be estimated,

$$ \mathfrak{N}^* = (\mu_{\alpha}^*, h_{\beta}^{\alpha,*}) \quad \text{for } \alpha, \beta \in M \cup P. $$

Finally, let $\Theta = \mathbb{R}^{M \cup P} \times L^2((0, A])^{(M \cup P)^2}$ the set of all possible candidates for $\mathfrak{N}^*$ so that $\mathfrak{N}^* \in \Theta$.

From equations (7.2) it is clear that there exist functions $\Psi^\alpha$ for $\alpha \in M \cup P$ such that

$$ \lambda_t^m = \Psi^m(\mathfrak{N}^*, t, X_{t-A,t}), \quad m \in M, t \in \mathbb{R}, $$

$$ W_k^p = \Psi^p(\mathfrak{N}^*, k, X_{k-A,k}) + \varepsilon_k^p, \quad p \in M, k \in D_p. $$

Given $F = (F^\alpha)_{\alpha \in M \cup P}$ where $F^m : \mathbb{R} \to \mathbb{R}$ and $F^p : D_p \to \mathbb{R}$ define

$$ (F \bullet X)_{[0,T)} = \sum_{m \in M} \int_{0}^{T-} F^m(t) dS_t^m + \sum_{p \in P} \int_{0}^{T-} F^p(k) W_k^p d\mathcal{D}_k^p. $$

We also define the following norm

$$ \|F\|_{[0,T)}^2 = \sum_{m \in M} \int_{0}^{T-} F^m(t)^2 dt + \sum_{p \in P} \int_{0}^{T-} F^p(k)^2 d\mathcal{D}_k^p, $$

and the associated scalar product $\langle \cdot, \cdot \rangle_{[0,T)}$.

Let $\mathfrak{N}, \mathfrak{N}' \in \Theta$, since by linearity we have $\|\Psi(\mathfrak{N}, X) - \Psi(\mathfrak{N}', X)\|_{[0,T)} = \|\Psi(\mathfrak{N} - \mathfrak{N}', X)\|_{[0,T)}$, we define the (pseudo and random) norm

$$ \|\mathfrak{N} - \mathfrak{N}'\|_{\Theta,T} = \|\Psi(\mathfrak{N}, X) - \Psi(\mathfrak{N}', X)\|_{[0,T)}. $$
With these notations we can introduce the following classical Least Square contrast [28, 47, 54].

**Definition 7.1** (Least square contrast). For $\mathfrak{N} \in \Theta$, we define the contrast by,

$$ \mathcal{C}(\mathfrak{N}) = -2 \sum_{m \in \mathcal{M}} \int_{0}^{T-} \Psi^{m}(\mathfrak{N}, t, X_{t-A,t}) dS_{t}^{m} + \sum_{m \in \mathcal{M}} \int_{0}^{T-} \Psi^{m}(\mathfrak{N}, t, X_{t-A,t})^{2} dt \\ -2 \sum_{p \in \mathcal{P}} \int_{0}^{T-} \Psi^{p}(\mathfrak{N}, k, X_{k-A,k}) W_{k}^{p} d\mathcal{D}_{k}^{p} + \sum_{p \in \mathcal{P}} \int_{0}^{T-} \Psi^{p}(\mathfrak{N}, k, X_{k-A,k})^{2} d\mathcal{D}_{k}^{p} $$

With the above notations we have,

$$ \mathcal{C}(\mathfrak{N}) = -2(\Psi(\mathfrak{N}, X) \bullet X)_{[0,T]} + \|\mathfrak{N}\|_{\Theta,T}^{2}. $$

**Remark 7.2.** The compensator of $\mathcal{C}(\mathfrak{N})$ is

$$ \tilde{\mathcal{C}}(\mathfrak{N}) = -2 \langle \Psi(\mathfrak{N}, X), \Psi(\mathfrak{N}^*, X) \rangle_{[0,T]} + \|\mathfrak{N}\|_{\Theta,T}^{2} \\ = \|\mathfrak{N} - \mathfrak{N}^*\|_{\Theta,T}^{2} - \|\mathfrak{N}^*\|_{\Theta,T}^{2}. $$

Therefore minimizing $\mathfrak{N} \mapsto \mathcal{C}(\mathfrak{N})$ makes sense since the compensator of the contrast is, up to an additive constant, the (non observable) quantity of interest to minimize.

As in [28], we introduce $\Phi$ a finite subset of $\Theta$, that we shall call a dictionary. In the sequel we only consider estimators among linear combinations of the dictionary, which means that any estimator will write as,

$$ \hat{\mathfrak{N}} = \sum_{\varphi \in \Phi} \hat{a}_{\varphi} \varphi := \mathfrak{N}_{\hat{a}}, \quad \hat{a} \in \mathbb{R}^{\Phi}. $$

The use of the dictionary $\Phi$ allows us to rewrite the contrast as a bilinear function of $a \in \mathbb{R}^{\Phi}$. Indeed since $\Psi$ is linear in $\mathfrak{N}$ we have

$$ \mathcal{C}(\mathfrak{N}_a) = -2a^\top b + a^\top G a $$

where $b \in \mathbb{R}^{\Phi}$ is a vector and $G \in \mathbb{R}^{\Phi^2}$ is a matrix with coordinates given by

$$ b_{\varphi} = (\Psi(\varphi, X) \bullet X)_{[0,T)}, \quad \text{and} \quad G_{\varphi, \varphi'} = \langle \Psi(\varphi, X), \Psi(\varphi', X) \rangle_{[0,T)}. $$

For later use, we also define $\tilde{b} \in \mathbb{R}^{\Phi}$ by

$$ \tilde{b}_{\varphi} = \langle \Psi(\varphi, X), \Psi(\mathfrak{N}^*, X) \rangle_{[0,T)}, \quad \varphi \in \Phi $$

so that $\tilde{\mathcal{C}}(\mathfrak{N}) = -2a^\top \tilde{b} + a^\top G a$.

The LASSO estimator consists minimizing the least square contrast plus an $L^1$ penalty. $L^1$ penalties are known to select variable, particularly when the dimension is large, as it may be the case here if $M$ or $P$ are large. As in [28] and [54], we allow the weights to depend on the data to adapt and fit the level of noise [60].

**Definition 7.3** (LASSO estimator). Let $\gamma > 0$ a tuning parameter and $d \in \mathbb{R}^{\Phi}$ non negative weights, that might depend on the observation $X_{-A,T}$. We define $\hat{a}$ by

$$ (7.3) \qquad \hat{a} \in \underset{a \in \mathbb{R}^{\Phi}}{\operatorname{argmin}} \{-2a^{\top}b + a^{\top}Ga + \gamma d^{\top}|a|\} $$

with $|a| = (|a_{\varphi}|)_{\varphi \in \Phi}$. We also define $\hat{\mathfrak{N}}$ the LASSO estimator of $\mathfrak{N}^*$ by

$$ (7.4) \qquad \hat{\mathfrak{N}} = \mathfrak{N}_{\hat{a}}. $$
The next step is to obtain theoretical guarantees on this estimator, and more precisely an oracle inequality.

To obtain oracle inequalities for the LASSO estimator we need two conditions. The first one is a control on the fluctuations of the contrast around its compensator. More precisely we have to control $b - \tilde{b}$. Given this control we can choose the weights $d$ in consequence and calibrate the LASSO [28, 60]. The second condition is about invertibility like properties of the Gram matrix $G$, see [57]. In [28] the authors went for a full invertibility property; the Gram matrix is invertible and its smallest eigenvalue is $\eta > 0$. Since requiring $G$ to be invertible may be restrictive, we turn instead on Restricted Eigenvalue (RE) properties which is a partial invertibility property as in [54]. RE property is a classical assumption in high dimensional LASSO procedure to prove oracle inequalities [57]. As opposed to [54], we prove, in Section 7.3.2 that the RE holds with high probability. This has also been done for discrete time Hawkes processes in [47] using Kalikow decomposition.

For a vector $a \in \mathbb{R}^\Phi$ and a subset of index $J \subset \Phi$, we define the vector $a_J \in \mathbb{R}^\Phi$ by $(a_J)_i = a_i 1_{i \in J}$.

**Definition 7.4 (Restricted Eigenvalue property).** Let $\eta > 0$, $c > 0$ and $s \in \mathbb{N}$. We say that the matrix $G$ satisfies the property $\text{RE}(\eta, c, s)$ if for all $a \in \mathbb{R}^\Phi$, for all subset $J \subset \Phi$ with $\text{Card}(J) \le s$ such that

$$|a_{Jc}|_1 \le c|a_J|_1,$$

then the following holds,

$$a^\top G a \ge \eta \|a_J\|^2.$$

**Remark 7.5.** If the dimension of $G$ is large, RE property allows to get invertibility properties of $G$ for vectors $a$ whose $L^1$ norms are mainly supported on $s$ coordinates.

We can now state an oracle inequality for the LASSO estimator.

**Theorem 7.6 (Oracle inequality).** Let $\gamma \ge 2$, $\eta > 0$ and $s \in \mathbb{N}$. Then, on the event on which

$$(i) \quad \forall \varphi \in \Phi, |b_{\varphi} - \tilde{b}_{\varphi}| \le d_{\varphi},$$

$$(ii) \quad G \text{ satisfies the property } \text{RE}(\eta, c(\gamma), s) \text{ with } c(\gamma) = \frac{\gamma+2}{\gamma-2},$$

the following inequality is satisfied for the LASSO estimator with tuning parameter $\gamma$ and weights $d$:

$$(7.5) \qquad \|\mathfrak{N} - \mathfrak{N}^*\|_{\Theta, T}^2 \le \inf_{a \in \mathbb{R}^\Phi, \ |\text{supp}(a)| \le s} \left\{ \|\mathfrak{N}_a - \mathfrak{N}^*\|_{\Theta, T}^2 + \eta^{-1} \left(\frac{\gamma+2}{2}\right)^2 \sum_{\varphi \in \text{supp}(a)} d_{\varphi}^2 \right\}.$$

**Remark 7.7.** Theorem 7.6 is only based on algebraic calculations on the definition of $\hat{a}$ and has no links with the HAR background. Indeed, proving Theorem 7.6 is equivalent to showing that, under conditions (i) and (ii), the following holds:

$$-2\hat{a}^\top \tilde{b} + \hat{a}^\top G \hat{a} \le \inf_{a \in \mathbb{R}^\Phi, \ |\text{supp}(a)| \le s} \left\{ -2a^\top \tilde{b} + a^\top G a + \eta^{-1} \left(\frac{\gamma+2}{2}\right)^2 \sum_{\varphi \in \text{supp}(a)} d_{\varphi}^2 \right\}.$$

We refer to the proof of Theorem 8 of [54] for the proof. If (i) and (ii) are satisfied with high probability then (7.5) holds also with high probability. Having theoretical lower bounds on the probability of events (i) and (ii), which are a bound on fluctuations around the compensator of the contrast and an RE property on $G$, requires complex probabilistic arguments on HAR processes. However, as long as conditions (i) and (ii) hold, inequality (7.5) holds and one can use it.

### 7.3. Lower bound on the Oracle event

This subsection is dedicated to prove lower bounds on the probability of events (i) and (ii) of Theorem 7.6. To do so we need more assumptions.

First we assume the noises to be gaussian: $\varepsilon_k^p \sim \mathcal{N}(0, \sigma_p^2)$. We also assume the standard deviations $\sigma_p$ to be known, or at least that we know a lower and an upper bound $\underline{\sigma}$ and $\overline{\sigma}$:

$$(7.6) \qquad \forall p \in \mathbf{P}, \ 0 < \underline{\sigma} \le \sigma_p \le \overline{\sigma} < \infty.$$
Then we also suppose that assumptions (Spec1) [M*], (Spec2) [M*,+] and (ExpTail) [M*] hold. Since the random drifts are gaussian, it is also clear that (FiMe) and (SubG) hold.

Lastly, we specify the typical architecture of the dictionary $\Phi$, more precisely we assume the following.

**Assumption (Dict).** The dictionary is elementary: each element $\varphi$ of the dictionary $\Phi$ has only one coordinate which is non zero, and $\Phi$ is bounded by one,

$$ \|\Phi\|_{\infty} \le 1 $$

where

$$ \|\Phi\|_{\infty} = \max_{\varphi \in \Phi} \max\{\|f\|_{\infty} \mid f \text{ coordinate of } \varphi\}. $$

Here $f$ is either a real number, in this case $\|f\|_{\infty} = |f|$, or a function, in this case $\|f\|_{\infty}$ is the supremum norm of $f$.

**Remark 7.8.** Having an elementary dictionary is not mandatory but it makes things easier. One can have a non elementary dictionary by adapting the weights $d$. However, a bounded dictionary is crucial. Without loss of generality, we can suppose that $\Phi$ is bounded by 1.

The above assumption allows us to decompose $\Phi$ as follows,

$$ \Phi = \bigcup_{\alpha \in M \cup P} \left( \Phi_{\alpha} \cup \bigcup_{\beta \in M \cup P} \Phi_{\beta}^{\alpha} \right) $$

where $\Phi_{\alpha}$ contains the elements $\varphi \in \Phi$ for which the only non zero coordiante is a base rate of type $\alpha$, ie $\varphi = \mu_{\alpha}$. Similarly, $\Phi_{\beta}^{\alpha}$ contains the elements $\varphi \in \Phi$ being an interaction function $h_{\beta}^{\alpha}$. Clearly, to avoid redundancy, we should have $\text{Card}(\Phi_{\alpha}) = 1$ for all $\alpha$. Let

$$ \Phi_M = \bigcup_{m \in M} \Phi_m, \quad \text{and} \quad \Phi^M = \bigcup_{m \in M, \alpha \in M \cup P} \Phi_{\alpha}^m $$

and similarly

$$ \Phi_P = \bigcup_{p \in P} \Phi_p, \quad \text{and} \quad \Phi^P = \bigcup_{p \in P, \alpha \in M \cup P} \Phi_{\alpha}^p. $$

We have divided the dictionary into four distinct subsets being, the point processes base rates, the chains base rates, the interaction functions on point processes and interaction functions on chains.

### 7.3.1. Bernstein inequalities to calibrate the LASSO

In this subsection we control the fluctuations of $b$ around $\tilde{b}$. For $\varphi \in \Phi$, let us write,

$$ b_{\varphi} - \tilde{b}_{\varphi} = J_{\varphi, M} + J_{\varphi, P} $$

where

$$ J_{\varphi, M} = \sum_{m \in M} \int_{0}^{T-} \Psi^{m}(\varphi, t, X_{t-A, t}) (dS_{t}^{m} - \lambda_{t}^{m} dt) $$

and

$$ J_{\varphi, P} = \sum_{p \in P} \int_{0}^{T-} \Psi^{p}(\varphi, k, X_{k-A, k}) \varepsilon_{k}^{p} d\mathcal{D}_{k}^{p}. $$

For $J_{\varphi,M}$ we use a Bernstein-type inequality for point processes given in Lemma E.2 in Appendix E. The second term of $b_{\varphi} - \tilde{b}_{\varphi}$ is the sum of a predictable process multiply by centered gaussian random variables. This term is controlled with, again, a Bernstein inequality adapted to this case stated in Lemma E.3 in Appendix E.

Combining the results of Lemmas E.2, E.3 leads to the following theorem that bounds the fluctuations of $b - \tilde{b}$ with high probability.
**Theorem 7.9 (Calibration of the LASSO).** There exists a constant $C^*$ depending only on $\mathfrak{M}^*$ such that for any $q > 0$ and any $\varphi \in \mathfrak{F}$, if

$$d_{\varphi} = \begin{cases} C^*(q+1) \log(T) \sqrt{T} & \text{if } \varphi \in \mathfrak{F}_M \\ C^*(q+1)^2 \log(T)^2 \sqrt{T} & \text{if } \varphi \in \mathfrak{F}_M \\ C^* \sqrt{(q+1) \log(T)} \sqrt{T} & \text{if } \varphi \in \mathfrak{F}_P \\ C^*(q+1)^{3/2} \log(T)^{3/2} \sqrt{T} & \text{if } \varphi \in \mathfrak{F}_P, \end{cases}$$

then we have,

$$\mathbb{P}(|b_{\varphi} - \tilde{b}_{\varphi}| \le d_{\varphi}) \ge 1 - \frac{C^*}{T^q}.$$

**Remark 7.10.** Depending on the nature of $\varphi$ we have different order of magnitude for $d_{\varphi}$. It is due to the fact that if for example $\varphi$ is a base rate then $\Psi(\varphi, X)$ is not random, whereas $\Psi(\varphi, X)$ is random if $\varphi$ is an interaction function. Also, it is harder to control the martingale $dS_t^m - \lambda_t^m dt$ than the gaussian noises $\varepsilon_k^p$ which explains the other differences between the four cases.

### 7.3.2. Restricted Eigenvalue property for the Gram matrix

The goal of this subsection is to prove that $G$ satisfies a RE property with high probability.

Recall that

$$G_{\varphi\varphi'} = \sum_{m \in \mathcal{M}} \int_{0}^{T-} \Psi^m(\varphi, t, X_{t-A,t}) \Psi^m(\varphi', t, X_{t-A,t}) dt + \sum_{p \in \mathcal{P}} \int_{0}^{T-} \Psi^p(\varphi, k, X_{k-A,k}) \Psi^p(\varphi', k, X_{k-A,k}) d\mathcal{D}_k^p$$

and define, with the notation of (12.2), the matrix $G_0^1$ by

$$(G_0^1)_{\varphi\varphi'} = \sum_{m \in \mathcal{M}} \int_0^{1-} \Psi^m(\varphi, t, X_{t-A,t}) \Psi^m(\varphi', t, X_{t-A,t}) dt + \sum_{p \in \mathcal{P}} \int_0^{1-} \Psi^p(\varphi, k, X_{k-A,k}) \Psi^p(\varphi', k, X_{k-A,k}) d\mathcal{D}_k^p.$$

The Restricted Eigenvalue property is stable under small perturbations in the following sense.

**Proposition 7.11.** Let $U$ and $V$ two square matrix and $\varepsilon > 0$. Suppose that $U \in \mathbb{RE}(\eta, c, s)$ and that

$$\forall i, j, |U_{ij} - V_{ij}| \le \varepsilon.$$

Then $V \in \mathbb{RE}(\eta - \varepsilon(c+1)^2s, c, s)$.

Our strategy is to use the above result with $V = G$ and $U = E[G]$, so that if $E[G]$ satisfies a RE property and if $G$ and $E[G]$ are close enough, then $G$ also satisfies a RE property. Thus we have to derive a concentration inequality for $G$, which is the purpose of Lemma 7.12 below.

**Lemma 7.12 (Concentration for $G$).** Let $T = 2n\tau + r$ with integers $n \ge 2A + 1$, $\tau > 0$ and $0 \le r < 2n$. There exist constants $C^*$ and $C_q^*$ depending only on $\mathfrak{M}$ and on $q$ for $C_q^*$ such that for all $q > 2$, for all $x > 0$,

$$\mathbb{P}(|G_{\varphi\varphi'} - E[G_{\varphi\varphi'}]| > T x + T C^* e^{-c^* n}) \le C^* \tau e^{-c^* n} + \frac{C_q^*}{\tau^{q-1}} \left( \frac{1}{x^q} + \frac{1}{x^{2(q-1)}} \right).$$

**Remark 7.13.** $G$ is the sum of $T$ variables distributed as $G_0^1$. To obtain concentration inequalities, classical results usually also require independence, which is not the case here. We use instead Berbee coupling, defined in Definition 6.5, to obtain concentration and prove Lemma 7.12, as explained in the proof section.

From stationarity, $E[G] = TE[G_0^1]$ and thus $E[G] \in \mathbb{RE}(T\eta, c, s) \Leftrightarrow E[G_0^1] \in \mathbb{RE}(\eta, c, s)$. This observation combined to the key Lemma 7.12, lead to the following theorem by taking $\tau \approx T^{1/2}$ and $n \approx T^{1/2}$.
**Theorem 7.14.** Let $T \in \mathbb{N}^*$, suppose that $\mathbb{E}[G_0^1] \in \mathbf{RE}(\eta_0, c, s)$ with $\eta_0 > 0$. Let $q > 1/2$ and $0 < \alpha < 1$. Then if

$$T \geq C^* \log^2 \left( \frac{C^*(c+1)^2 s}{\alpha \eta_0} \right), \quad T \geq 8(A+1)^2,$$

with probability at least

$$1 - \mathrm{Card}(\mathbf{\Phi})^2 \frac{C_q^*}{T^q} \left[ 1 + \left( \frac{(c+1)^2 s}{\alpha \eta_0} \right)^{4q} \right]$$

we have

$$G \in \mathbf{RE}((1 - \alpha)\eta_0 T, c, s)$$

where $C^*, C_q^*, C_{\gamma,q}^*$ are constants depending on $\mathfrak{M}^*$, $q$ for the last two and $\gamma$ for the last one.

In the previous result, we still need the assumption that $\mathbb{E}[G_0^1] \in \mathbf{RE}(\eta_0, c, s)$ with $\eta_0 > 0$. However, it is not clear at all how to compute $\mathbb{E}[G_0^1]$ in order to check if the RE condition holds. Indeed, we do not know the true parameters and even if we knew them, we have no expression for the distribution (to compute the expectation) in terms of the true parameters. To bypass this difficulty, define the probability $\mathbb{Q}$ under which $(S^m)_{m \in M}$ are independent homogeneous Poisson point processes of rates 1 after time $-A$ and the chain $W_k^p$ are independent $\mathcal{N}(0, \sigma_p^2)$ for $(p, k) \in \mathbb{Q}$ with $k \geq -A$.

Then the following result links $\mathbb{E}[G_0^1]$ to $\mathbb{E}_{\mathbb{Q}}[G_0^1]$ which is computable and depends only on $\mathbf{\Phi}$.

**Proposition 7.15.** Suppose that $\min_{m \in M} \mu_m > 0$ and that $h_\beta^\alpha$ are bounded for all $\alpha, \beta \in \mathbf{M} \cup \mathbf{P}$. Then there exists constants $C^*$ and $c^*$ depending only on $\mathfrak{M}^*$ such that, if $\mathbb{E}_{\mathbb{Q}}[G_0^1] \in \mathbf{RE}(\eta, c, s)$, we have, for all integer $T > 0$, for all $B \geq 1$,

$$\mathbb{E}[G_0^T] \in \mathbf{RE}(T e^{-C^* B^2} (\eta - C^*(c+1)^2 s e^{-c^* B}), c, s).$$

Therefore we have

$$\exists \eta_1 > 0, \mathbb{E}_{\mathbb{Q}}[G_0^1] \in \mathbf{RE}(\eta_1, c, s) \implies \exists \eta_0 > 0, \mathbb{E}[G_0^T] \in \mathbf{RE}(T \eta_0, c, s)$$

and for all $\alpha \in [0, 1]$, we can chose

$$\eta_0 \geq \eta_1 (1 - \alpha) \exp \left[ - C^* \left( \log_+ \left[ \frac{C^*(c+1)^2 s}{\alpha \eta_1} \right] \right)^2 \right].$$

With this result in hand, one can compute $\mathbb{E}_{\mathbb{Q}}[G_0^1]$ and check whether or not the RE condition holds. The choice of the dictionary is crucial since $\mathbb{E}_{\mathbb{Q}}[G_0^1]$ depends only on $\mathbf{\Phi}$. If the RE condition is satisfied, then one can apply Theorem 7.14 and has a control on the event under which the oracle inequality is stated.

# 8. Proof of results of Section 3

## 8.1. Proof of Theorem 3.4

The proof of Theorem 3.4 consists of showing that the naive way of constructing the process does not fail. This way of constructing the process can be resumed as follows

**Step 0:** For $t < t_0$ the process is given by the initial condition $\mathcal{C}$.

**Step 1:** Calculate the value of the process at $t = t_0$.

**Step 2:** If the process is constructed on $[t_0, t_1]$ then calculate the *temporary pre-intensities* at level $t_1$ for $t_1 < t \leq \min\{k \in \cup_{p \in \mathbf{P}} \mathcal{D}_p | k > t_1\} =: \mathcal{D}(t_1)$ defined by

$$\mathrm{Temp}_{t_1}(\ell^m)_t = \mu_m + \sum_{m'} \int_{-\infty}^{t_1} h_{m'}^m(t-s) dS_s^{m'} + \sum_{p \in \mathbf{P}} \int_{-\infty}^{t_1} h_p^m(t-k) \kappa_p^m (W_k^p) d\mathcal{D}_k^p.$$

The *temporary intensity* is defined by

$$\mathrm{Temp}_{t_1}(\lambda^m)_t = \Phi^m(\mathrm{Temp}_{t_1}(\ell^m)_t).$$
Basically, $\mathrm{Temp}_{t_1}(\lambda^m)_t$ (resp. $\mathrm{Temp}_{t_1}(\ell^m)_t$) is the intensity (resp. pre-intensity) of node $m$ at time $t$ if there are no points (from any node) between $t_1$ and $t$. Then one have access to the pre-points at level $t_1$, being the atoms $(s, x)$ of the Poisson random measures $\pi^m$ such that $t_1 < s \le \mathcal{D}(t_1)$ and $0 \le x \le \mathrm{Temp}_{t_1}(\lambda^m)_t$.

**Step 3:** If there are no pre-points at level $t_1$ it means that there is no points in $(t_1, \mathcal{D}(t_1)]$. In this case one can calculate with (2.2c) the chains $W_k^p$ for which $k = \mathcal{D}(t_1)$ using what has already been constructed. The process is now fully constructed on $[t_0, \mathcal{D}(t_1)]$ and one can apply Step 2 with $t_1 \leftarrow \mathcal{D}(t_1)$. If there are pre-points then one has to find the first one: the one with minimal time coordinate, if it exists. Let say this time coordinate is $t_2$. Then this pre-point with time coordinate $t_2$ is in fact a true point and the process is fully construct on $[t_0, t_2]$ and one can apply Step 2 with $t_1 \leftarrow t_2$.

**Proof of Theorem 3.4.** The construction explained above may encounter two different problems, 1. it is possible that after an infinite number of steps it only constructs the process on a finite time interval, 2. it is also possible that there is no pre-point with minimal time coordinate in step 3, it might happen if there is an accumulation of pre-points. We will show that none of these happen.

Let $t^* > 0$ be small enough (the precise condition will be given at the end of the proof). Let

$$ \{t_0 < t_1 < t_2 < \dots\} = (t^* \mathbb{N} + t_0) \cup \bigcup_{p \in \mathcal{P}} \mathcal{D}_p \cap [t_0, \infty). $$

With this definition we have $t_n \to \infty$ as $n \to \infty$. We will show by induction on $n$ that the total number of pre-points, with time coordinate in $[t_0, t_n]$ we will encounter through the construction is almost surely finite and that almost surely, for any $(p, k) \in \mathcal{Q}$ with $t_0 \le k \le t_n$ we have $|w_k^p| < \infty$. Thus the construction has to reach $t_n$ in a finite number of steps and cannot fail. It will conclude the existence part of the proof since all true points are, at some point, pre-points, so non explosion will be clear.

Let us first explain how an integrable initial condition relates to assumption (NoExpl). Let $f \in L^1$ a non negative function then we have

$$ \begin{aligned} \int_a^b \int_{-\infty}^t f(t-s)d\mathcal{E}_s^m dt &= \int_{-\infty}^b \int_{a \wedge s}^b f(t-s)dt d\mathcal{E}_s^m \\ &\le \int_{-\infty}^b \|f\|_{1, [a-s, b-s]} d\mathcal{E}_s^m \\ &\le \sum_{n \ge 0} \|f\|_{1, [a-b+n, n+1]} \mathrm{Card}(\mathcal{E}^m \cap [b-(n+1), b-n]). \end{aligned} $$

Thus is $\mathcal{E}$ is an integrable initial condition,

$$ (8.1) \qquad \mathrm{E} \left[ \int_a^b \int_{-\infty}^t f(t-s) d\mathcal{E}_s^m dt \right] \le K(a-b+2) \|f\|_1 < \infty. $$

As a consequence, $\int_a^b \int_{-\infty}^t f(t-s)d\mathcal{E}_s^m dt < \infty$ almost surely and $\int_{-\infty}^t f(t-s)d\mathcal{E}_s^m < \infty$ almost surely almost everywhere.

Similarly, for any non negative $L^1$ function $f$ and non negative Lipschitz function $g$, we have

$$ \begin{aligned} \int_a^b \int_{-\infty}^t f(t-k)g(\mathcal{E}_k^p) d\mathcal{D}_k^p dt &= \int_{-\infty}^b \int_{a \wedge k}^b f(t-k)g(\mathcal{E}_k^p) dt d\mathcal{D}_k^p \\ &\le \int_{-\infty}^b \|f\|_{1, [a-k, b-k]} g(\mathcal{E}_k^p) d\mathcal{D}_k^p \\ &\le \int_{-\infty}^b \|f\|_{1, [a-k, b-k]} [\mathrm{Lip}(g)|\mathcal{E}_k^p| + g(0)] \mathbb{1}_{\mathcal{E}_k^p \ne \emptyset} d\mathcal{D}_k^p. \end{aligned} $$
Thus for integrable initial condition,

$$ (8.2) \qquad \mathbb{E} \left[ \int_a^b \int_{-\infty}^t f(t-k)g(\mathfrak{c}_k^p) d\mathcal{D}_k^p dt \right] < \infty. $$

With these results in hand, let us now prove the induction.

**n=0:** Almost surely there is no pre-points in the time interval $[t_0, t_0]$ since almost surely the Poisson random measure have no atoms lying on a given line. If $t_0 = k$ with $(p, k) \in \mathcal{Q}$ then one can calculate $W_k^p$ with (2.2c) and the initial condition. In this case we have $|W_k^p| < \infty$ almost surely by (8.1) and (8.2).

**n→n+1:** The process is constructed on $[t_0, t_n]$ and by induction, the total number of points before in $[t_0, t_n]$ is $\mathcal{F}_{t_n}$ almost surely finite and all the $W_k^p$ with $t_0 \le k \le t_n$ are also almost surely finite for $\mathcal{F}_{t_n}$. We can calculate the temporary pre-intensities at level $t_n$ for $t_n < t \le t_{n+1}$,

$$ \mathrm{Temp}_{t_n}(\ell^m)_t = \mu_m + \sum_{m'} \int_{-\infty}^{t_n} h_{m'}^m(t-s) dS_s^{m'} + \sum_{p \in \mathcal{P}} \int_{-\infty}^{t_n} h_p^m(t-k) \kappa_p^m(W_k^p) d\mathcal{D}_k^p. $$

We denote by $E_0$ the set of all pre-points at level $t_n =: s_0$, see Figure 4. Using the induction hypothesis, (NoExpl), (8.1) and (8.2) we obtain that

$$ \begin{aligned} & \int_{t_n}^{t_{n+1}} |\mathrm{Temp}_{t_n}(\ell^m)_s| ds \le t^* |\mu_m| + \sum_{m'} \int_{t_n}^{t_{n+1}} \int_{-\infty}^{t_n} |h_{m'}^m(t-s)| dS_s^{m'} dt \\ & \qquad + \sum_{p \in \mathcal{P}} \int_{t_n}^{t_{n+1}} \int_{-\infty}^{t_n} |h_p^m(t-k)| |\kappa_p^m(W_k^p)| d\mathcal{D}_k^p dt \\ & < \infty \quad \mathcal{F}_{t_n} - \mathrm{a.s.} \end{aligned} $$

Since from (NoExpl) the $\Phi^m$ are Lipschitz, it implies that

$$ \mathbb{E}[\mathrm{Card}(E_0)|\mathcal{F}_{t_n}] = \sum_m \int_{t_n}^{t_{n+1}} \mathrm{Temp}_{t_n}(\lambda^m)_s ds < \infty \quad \mathrm{a.s.}, $$

and so, $\mathcal{F}_{t_n}$-almost surely, $\mathrm{Card}(E_0) < \infty$.

Of course the set $E_0$ does not contain, in general, all the pre-points we are going to encounter in $[t_n, t_{n+1}]$ through the construction. Indeed, if $E_0 \neq \emptyset$, we will apply again Step 2 of the construction at time $s_1$: the time of the first pre-point in $E_0$, then calculate temporary intensities at level $s_1$, then have a set $E_1$ of pre-points at level $s_1$ which might contains new pre-points: pre-points at level $s_1$ that weren't in $E_0$, ie that they weren't pre-points at level $t_n = s_0$, see Figure 4. And so on, we apply Step 2 at time $s_2$ being the time of the first pre-point in $E_1$, calculate temporary intensities at level $s_2$, get the set $E_2$ of pre-points at level $s_2$ and maybe find some new pre-points: pre-points at level $s_2$ but not at level $s_1$ or $s_0$. The key of the reasoning is to stochastically dominate the number of new pre-points after each time we run Step 2 of the construction, ie the number of pre-points at level $s_k$ but not at level $s_{k-1}, \dots, s_0$, that is $\mathrm{Card}(E_k \setminus (E_0 \cup \dots \cup E_{k-1}))$. Each time Step 2 is run, the number of new pre-points follows a Poisson distribution of parameter being the sum, for each node $m$, of the area between $\mathrm{Temp}_{s_k}(\lambda^m)_t$ and $\max_{i=0, \dots, k-1} \mathrm{Temp}_{s_i}(\lambda^m)_t$ when the first one is above the second one and $t \in ]s_k, t_{n+1}]$. For $k=1$, it is the sum for each node of the shaded zone area in Figure 4.

Suppose that the pre-point which starts Step 2 is from node $m_0$ and appears at time $s_k \in (t_n, t_{n+1}]$. Then we have,

$$ \begin{aligned} \left( \mathrm{Temp}_{s_k}(\lambda^m)_t - \max_{i=0, \dots, k-1} \mathrm{Temp}_{s_i}(\lambda^m)_t \right)_+ &= \left( \Phi^m(\mathrm{Temp}_{s_k}(\ell^m)_t) - \max_{i=0, \dots, k-1} \Phi^m(\mathrm{Temp}_{s_i}(\ell^m)_t) \right)_+ \\ &\le \left( \Phi^m(\mathrm{Temp}_{s_k}(\ell^m)_t) - \Phi^m(\mathrm{Temp}_{s_{k-1}}(\ell^m)_t) \right)_+ \\ &\le L_m |\mathrm{Temp}_{s_k}(\ell^m)_t - \mathrm{Temp}_{s_{k-1}}(\ell^m)_t| \\ &= L_m |h_{m_0}^m(t - s_k)|. \end{aligned} $$
Initialisation :

$$
\left\{
\begin{aligned}
\lambda_t^{m,0} &= \Phi^m(\mu_m) \\
S^{m,0}(C) &= \int_C \int_{\mathbb{R}_+} 1_{x \le \lambda_t^{m,0}} d\pi^m(t,x) \\
W_k^{p,0} &= \xi_k^p
\end{aligned}
\right.
$$

Iteration $n \to n+1$ :

$$
\left\{
\begin{aligned}
\lambda_t^{m,n+1} &= \Phi^m \left( \mu_m + \sum_{m' \in \mathcal{M}} \int_{-\infty}^{t-} h_{m'}^m (t-s) dS_s^{m',n} + \sum_{p \in \mathcal{P}} \int_{-\infty}^{t-} h_p^m (t-k) \kappa_p^m (W_k^{p,n}) d\mathcal{D}_k^p \right) \\
S^{m,n+1}(C) &= \int_C \int_{\mathbb{R}_+} 1_{x \le \lambda_t^{m,n+1}} d\pi^m(t,x) \\
W_k^{p,n+1} &= \xi_k^p + \Phi^p \left( \sum_{m \in \mathcal{M}} \int_{-\infty}^{k-} h_m^p (k-s) dS_s^{m,n} + \sum_{p' \in \mathcal{P}} \int_{-\infty}^{k-} h_{p'}^p (k-k') \kappa_{p'}^p (W_{k'}^{p',n}) d\mathcal{D}_{k'}^{p'} \right).
\end{aligned}
\right.
$$

Whenever one of the integral inside the link functions does not converge, we arbitrarily set its value to 0. As we will see,
it essentially never happens.

**Proof of Theorem 3.9.** The proof can be divided into 4 steps. 1) Checking that the Picard iteration is well defined, 2) Proving that the Picard iteration converges almost surely and in $L^1$, 3) Showing that the limit is a solution to the HAR equations, 4) Showing uniqueness.

Let for $n \ge 0$,

$$
\begin{align*}
\Lambda_n^m &= \sup_{t \in \mathbb{R}} \mathbb{E} [|\lambda_t^{m,n+1} - \lambda_t^{m,n}|] \\
\Gamma_n^p &= \sup_{k \in \mathcal{D}_p} \mathbb{E} [|W_k^{p,n+1} - W_k^{p,n}|] \\
\Psi_n &= \max \left( \sup_{m \in \mathcal{M}, t \in \mathbb{R}} \mathbb{E} [|\lambda_t^{m,n}|], \sup_{(p,k) \in \mathcal{Q}} \mathbb{E} [|W_k^{p,n}|] \right).
\end{align*}
$$

**Step 1:** One can easily see that $\Psi_0 < \infty$, it follows from assumption (FiMe) and the fact that $\lambda_t^{m,0} = \Phi^m(\mu_m)$ is
deterministic. Thus we have

$$
\begin{align*}
\mathbb{E} \left[ \int_{-\infty}^{t-} |h_{m'}^m (t-s) | dS_s^{m',0} \right] &= \mathbb{E} \left[ \int_{-\infty}^{t-} |h_{m'}^m (t-s) | \lambda_s^{m',0} ds \right] \\
&\le \Psi_0 \int_{-\infty}^{t-} |h_{m'}^m (t-s) | ds < \infty.
\end{align*}
$$

Thus, almost surely $\int_{-\infty}^{t-} h_{m'}^m (t-s) dS_s^{m',0}$ is well defined. Similarly, it is the case for any other integral involved in
the definition of $\lambda_t^{m,1}$ or $W_k^{p,1}$. Thus, $W_k^{p,1}$ is well defined almost surely and $\lambda^{m,1}$ is predictable and well defined
d$\mathbb{P}$ ($\omega$) $\otimes$ $dt$ almost everywhere, which implies that $S^{m,1}$ is well defined almost surely. Finally, one easily check that
$\Psi_1 \le K(1+\Psi_0) < \infty$ with K a constant depending on $\mathfrak{M}$. Then an induction can be performed by applying the same
reasoning and ensures that the Picard iteration is well defined and that $\Psi_n < \infty$ for all $n \in \mathbb{N}$.
**Step 2:** The goal of this step is to show that

$$ \forall m \in \mathbf{M}, \forall t \in \mathbb{R}, \lambda_t^{m,n} \xrightarrow{n \to \infty} \lambda_t^m \in \mathbb{R}, $$

$$ \forall (p, k) \in \mathcal{Q}, W_k^{p,n} \xrightarrow{n \to \infty} W_k^p \in \mathbb{R}, $$

$$ \sup_{m \in \mathcal{M}, t \in \mathbb{R}} \mathbb{E}[|\lambda_t^m|] < \infty \quad \text{and} \quad \sup_{(p,k) \in \mathcal{Q}} \mathbb{E}[|W_k^p|] < \infty, $$

$$ \forall m, \forall C \subset \mathbb{R} \text{ bounded, } S^{m,n}(C) \xrightarrow{n \to \infty} S^m(C) \in \mathbb{N}, $$

The almost sure event of convergence can be chosen uniform over all $(p, k) \in \mathcal{Q}$ and all $C \subset \mathbb{R}$ bounded.

Let $t \in \mathbb{R}$ and $n \in \mathbb{N}^*$, since $\Phi^m$ is $L_m$ Lipschitz,

$$ |\lambda_t^{m,n+1} - \lambda_t^{m,n}| \le L_m \left| \sum_{m'} \int_{-\infty}^{t-} h_{m'}^m(t-s) (dS_s^{m',n} - dS_s^{m',n-1}) \right. \\ \left. \qquad + \sum_{p \in \mathcal{P}} \int_{-\infty}^{t-} h_p^m(t-k) (\kappa_p^m(W_k^{p,n}) - \kappa_p^m(W_k^{p,n-1})) d\mathcal{D}_k^p \right| $$

We can rewrite the first integral in the RHS as follows,

$$ \sum_{m'} \int_{-\infty}^{t-} \int_{\mathbb{R}_+} h_{m'}^m(t-s) \left( \mathbb{1}_{\lambda_s^{m',n-1} < x \le \lambda_s^{m',n}} - \mathbb{1}_{\lambda_s^{m',n} < x \le \lambda_s^{m',n-1}} \right) d\pi^{m'}(s,x). $$

Noticing that

$$ |\mathbb{1}_{\lambda_s^{m',n-1} < x \le \lambda_s^{m',n}} - \mathbb{1}_{\lambda_s^{m',n} < x \le \lambda_s^{m',n-1}}| = \mathbb{1}_{\lambda_s^{m',n-1} \wedge \lambda_s^{m',n} < x \le \lambda_s^{m',n-1} \vee \lambda_s^{m',n}} $$

it follows that,

$$ \begin{aligned} \mathbb{E}[|\lambda_t^{m,n+1} - \lambda_t^{m,n}|] & \le \sum_{m'} \int_{-\infty}^{t} L_m |h_{m'}^m(t-s)| \mathbb{E}[|\lambda_s^{m',n} - \lambda_s^{m',n-1}|] ds \\ & \qquad + \sum_{p \in \mathcal{P}} \int_{-\infty}^{t-} L_m |h_p^m(t-k)| \mathbb{E}[|\kappa_p^m(W_k^{p,n}) - \kappa_p^m(W_k^{p,n-1})|] d\mathcal{D}_k^p. \\ & \le \sum_{m'} L_m \|h_{m'}^m\|_1 \Lambda_{n-1}^{m'} + \sum_{p \in \mathcal{P}} a_p^m L_m \|h_p^m\|_{1,n_p}^\infty \Gamma_{n-1}^p \end{aligned} $$

Thus we obtain,

$$ \Lambda_n^m \le \sum_{m'} L_m \|h_{m'}^m\|_1 \Lambda_{n-1}^{m'} + \sum_{p \in \mathcal{P}} a_p^m L_m \|h_p^m\|_{1,n_p}^\infty \Gamma_{n-1}^p. $$

Similarly, one can show that

$$ \Gamma_n^p \le \sum_m L_p \|h_m^p\|_1 \Lambda_{n-1}^m + \sum_{p' \in \mathcal{P}} a_{p'}^p L_p \|h_{p'}^p\|_{1,n_{p'}}^\infty \Gamma_{n-1}^{p'} $$

Let $\Lambda_n = (\Lambda_n^m)_{m \in \mathcal{M}}$ and similarly $\Gamma_n = (\Gamma_n^p)_{p \in \mathcal{P}}$. The two previous equations rewrites

$$ \left( \begin{array}{c} \Lambda_n \\ \Gamma_n \end{array} \right) \le H_{\mathfrak{M}} \left( \begin{array}{c} \Lambda_{n-1} \\ \Gamma_{n-1} \end{array} \right). $$
Iterating gives immediately

$$ \left( \frac{\Lambda_n}{\Gamma_n} \right) \leq (H_{\mathfrak{M}})^n \left( \frac{\Lambda_0}{\Gamma_0} \right). $$

Since we have

$$ \Lambda_0^m, \Gamma_0^p \leq \Psi_0 + \Psi_1 < \infty, $$

it follows from assumption (Spec1)[M] that there exist constants $\text{SpR}(H_{\mathfrak{M}}) \le r < 1$ and $K > 0$ such that, for all $m \in M, p \in P$,

$$ (8.3) \qquad \forall n \in \mathbb{N}, \Lambda_n^m, \Gamma_n^p \leq Kr^n. $$

Since the $0 \le r < 1$, the series $\sum_n \Lambda_n^m$ converges, and so,

$$ \sum_n \mathbb{E} \left[ \left| \lambda_t^{m,n+1} - \lambda_t^{m,n} \right| \right] < \infty $$

which is the same as

$$ \mathbb{E} \left[ \sum_n \left| \lambda_t^{m,n+1} - \lambda_t^{m,n} \right| \right] < \infty. $$

Therefore, almost surely,

$$ \sum_n |\lambda_t^{m,n+1} - \lambda_t^{m,n}| < \infty $$

which implies that for all $t \in \mathbb{R}$,

$$ \lambda_t^{m,n} \xrightarrow{n \to \infty} \lambda_t^m \in \mathbb{R}. $$

And the same reasoning gives,

$$ \forall (p,k) \in \mathcal{Q}, W_k^{p,n} \xrightarrow{n \to \infty} W_k^p \in \mathbb{R}. $$

Since $\mathcal{Q}$ is countable, the almost sure event of convergence can be taken uniformly over $\mathcal{Q}$.

From (8.3) and $\Psi_0 < \infty$ the $L^1$ bounds and $L^1$ convergences are clear.

Let us denote by $\triangle$ the symmetric subtraction $(E \triangle F = (E \setminus F) \cup (F \setminus E)$ for $E, F$ sets). Let $j \in \mathbb{N}^*$ and $m \in M$, then we have

$$ \begin{aligned} \mathbb{E} [\text{Card}((S^{m,n} \triangle S^{m,n+1}) \cap [-j,j])] &= \int_{[-j,j]} \mathbb{E} [\left| \lambda_s^{m,n} - \lambda_s^{m,n+1} \right|] ds \\ &\le 2j \Lambda_n^m. \end{aligned} $$

Since the cardinal only takes non negative integer values and because $\sum_n 2j\Lambda_n^m < \infty$, by Borel-Cantelli Lemma, almost surely (uniformly in $j \in \mathbb{N}^*$ since the integers are countable), there exists $n_0 = n_0(j, \omega)$ such that for all $n \in \mathbb{N}$, for all $m \in M$,

$$ S^{m,n_0+n} \cap [-j,j] = S^{m,n_0} \cap [-j,j]. $$

Then, we can define $S^m$ by $S^m \cap [-j,j] = S^{m,n_0(j)} \cap [-j,j]$ for all $j$.

Finally, if $C \subset \mathbb{R}$ is bounded, then there exists $j \in \mathbb{N}^*$ such that $C \subset [-j,j]$, thus, almost surely (which does not depend on $C$), we have

$$ S^{m,n}(C) = \text{Card} (S^{m,n} \cap [-j,j] \cap C) \xrightarrow{n \to \infty} \text{Card} (S^m \cap [-j,j] \cap C) =: S^m(C). $$

**Step 3:** In the this step we show that the limit process found in the Step 2 is indeed a $\text{HAR}_{\mathfrak{M}}(-\infty)$ process and is also 1-stationary.
First we prove that for all $m \in \mathcal{M}$, the set $S^m$ matches its aimed value $\bar{S}^m$ being the points under $\lambda^m$. One can observe that $\lambda^m$ is predictable as limit of predictable terms. So, for $j \in \mathbb{N}^*$ and $m \in \mathcal{M}$, we have

$$
\begin{align*}
\mathbb{E}[\text{Card}((S^{m,n} \triangle \bar{S}^m) \cap [-j, j))] &= \mathbb{E}\left[\int_{-j}^{j} \int_{\mathbb{R}_+} \mathbb{1}_{\lambda_s^{m,n} \wedge \lambda_s^m < x \le \lambda_s^{m,n} \vee \lambda_s^m} d\pi^m(s, x)\right] \\
&= \int_{-j}^{j} \mathbb{E}[|\lambda_s^{m,n} - \lambda_s^m|] ds \\
&\le 2j \frac{Kr^n}{1-r} \xrightarrow{n \to \infty} 0.
\end{align*}
$$

Borel-Cantelli Lemma applies once again and we conclude that

$$
a.s., \forall j \in \mathbb{N}^*, S^m \cap [-j, j] = \bar{S}^m \cap [-j, j]
$$

which implies that almost surely, $S^m = \bar{S}^m$.

Let us denote $\bar{\lambda}_t^m := \Phi^m \left( \mu_m + \sum_{m' \in \mathcal{M}} \int_{-\infty}^{t-} h_{m'}^m (t-s) dS_s^{m'} + \sum_{p \in \mathcal{P}} \int_{-\infty}^{t-} h_p^m (t-k) \kappa_p^m (W_k^p) d\mathcal{D}_k^p \right)$. Let us now check that for all $m \in \mathcal{M}$ we have

$$
\lambda_t^m = \bar{\lambda}_t^m \quad d\mathbb{P}(\omega) \otimes dt \quad \text{a.e.}
$$

Let $m \in \mathcal{M}$ and $t \in \mathbb{R}$, then,

$$
\begin{align*}
\mathbb{E}[|\lambda_t^{m,n} - \bar{\lambda}_t^m|] &\le L_m \left( \sum_{m' \in \mathcal{M}} \int_{-\infty}^{t-} |h_{m'}^m(t-s)| \mathbb{E}[|\lambda_s^{m',n} - \lambda_s^{m'}|] ds + \sum_{p \in \mathcal{P}} \int_{-\infty}^{t-} |h_p^m(t-k)| a_p^m \mathbb{E}[|W_k^{p,n} - W_k^p|] d\mathcal{D}_k^p \right) \\
&\le C r^n \xrightarrow{n \to \infty} 0
\end{align*}
$$

with $C$ a constant depending on $\mathfrak{M}$. Thus, for all $t \in \mathbb{R}$ we have $\mathbb{P}(\lambda_t^m = \bar{\lambda}_t^m) = 1$. Then, Fubini theorem implies that $d\mathbb{P}(\omega) \otimes dt$ almost everywhere, $\lambda_t^m = \bar{\lambda}_t^m$. Similarly, almost surely, for all $(p, k) \in \mathcal{Q}$,

$$
W_k^p = \xi_k^p + \Phi^p \left( \sum_m \int_{-\infty}^{k-} h_m^p (k-s) dS_s^m + \sum_{p' \in \mathcal{P}} \int_{-\infty}^{k-} h_{p'}^p (k-k') \kappa_{p'}^p (W_{k'}^{p'}) d\mathcal{D}_{k'}^{p'} \right).
$$

In the end we constructed a $\mathbf{HAR}_{\mathfrak{M}}(-\infty)$ process. It is also 1-stationary since in the Picard iteration the initial condition is 1-stationary and the evolution (through the Picard iteration) too, thus, the limit is also 1-stationary.

**Step 4:** In this last step we show uniqueness. Consider two solutions, indexed by 1 and 2, and define

$$
\beta^m = \sup_{t \in \mathbb{R}} \mathbb{E} [|(\lambda_t^m)_1 - (\lambda_t^m)_2|] < \infty
$$

and

$$
\omega^p = \sup_{k \in \mathcal{D}_p} \mathbb{E} [|(W_k^p)_1 - (W_k^p)_2|] < \infty.
$$

Then, using the equations (2.2), one has,

$$
\mathbb{E}[|(\lambda_t^m)_1 - (\lambda_t^m)_2|] \le \sum_{m'} \int_{-\infty}^{t-} L_m |h_{m'}^m(t-s)| \beta^{m'} ds + \sum_{p \in \mathcal{P}} \int_{-\infty}^{t-} L_m |h_p^m(t-k)| a_p^m \omega^p d\mathcal{D}_k^p.
$$

And thus,

$$
(8.4) \qquad \beta^m \le \sum_{m'} L_m \|h_{m'}^m\|_1 \beta^{m'} + \sum_{p \in \mathcal{P}} a_p^m L_m \|h_p^m\|_{1_{n_p}}^\infty \omega^p.
$$
Similarly,

$$ (8.5) \qquad \omega^p \le \sum_m L_p \|h_m^p\|_1 \beta^m + \sum_{p' \in \mathcal{P}} a_{p'}^p L_p \|h_{p'}^p\|_{1, n_{p'}}^{\infty} \omega^{p'}. $$

Thus if $\beta = (\beta^m)_{m \in \mathcal{M}}$ and $\omega = (\omega^p)_{p \in \mathcal{P}}$, we have

$$ \binom{\beta}{\omega} \le H_{\mathfrak{M}} \binom{\beta}{\omega}. $$

Iterating this formula gives for all $n \in \mathbb{N}^*$,

$$ \binom{\beta}{\omega} \le (H_{\mathfrak{M}})^n \binom{\beta}{\omega}. $$

And thus $\beta = 0$ and $\omega = 0$. Since $\mathcal{Q}$ is countable, a.s., $\forall (p, k) \in \mathcal{Q}$, $(W_k^p)_1 = (W_k^p)_2$. It also implies that $d\mathbb{P}(\omega) \otimes dt$ almost everywhere, $(\lambda_t^m)_1 = (\lambda_t^m)_2$. Finally, one also has,

$$ \begin{aligned} \mathbb{E} [\mathrm{Card} (S_1^m \triangle S_2^m)] &= \mathbb{E} \left[ \int_{\mathbb{R}} |(\lambda_t^m)_1 - (\lambda_t^m)_2| dt \right] \\ &\le \int_{\mathbb{R}} \beta^m dt \\ &= 0. \end{aligned} $$

Thus, a.s., $\forall m$, $S_1^m = S_2^m$.

□

# 9. Proof of results of Section 4

## 9.1. Proof of Lemma 4.3 and Proposition 4.4

**Proof of Lemma 4.3.** Given a sequence $Z$ it is clear that the sequence $V$ is well defined by the (4.1). Unrolling (4.1) gives an expression of $V_k^p$ as a fixed linear combination of the $Z_{k'}^{p'}$ with $k' \le k$, which proves existence of the coefficients. Uniqueness is obtained by setting the sequence $Z$ to $(1_{(p,k)=(p_0,k_0)})(p,k) \in \mathcal{Q}, k \ge k_0$ since then associated $V$ sequence satisfies

$$ V_k^p = \sum_{p' \in \mathcal{P}} \int_{k_0}^{k} \binom{p, k}{p', k'}_{\mathfrak{M}} Z_{k'}^{p'} d\mathcal{D}_{k'}^{p'} = \binom{p, k}{p_0, k_0}_{\mathfrak{M}} $$

which proves uniqueness in **point 1** since $V$ is well defined and forces the value of the coefficient.

We now prove **point 3**. The fact that $\langle p_1, k_0 \rangle_{\mathfrak{M}} = 1_{p_0=p_1}$ is quite clear, take $Z = (1_{(p,k)=(p_0,k_0)})(p,k) \in \mathcal{Q}, k \ge k_0$. On one side, by **point 1** we have

$$ \begin{aligned} V_k^p &= \sum_{p' \in \mathcal{P}} \int_{k_0}^{k} \binom{p, k}{p', k'}_{\mathfrak{M}} Z_{k'}^{p'} d\mathcal{D}_{k'}^{p'} \\ &= \binom{p, k}{p_0, k_0}_{\mathfrak{M}}, \end{aligned} $$

thus we have

$$ V_{k_0}^{p_1} = \binom{p_1, k_0}{p_0, k_0}_{\mathfrak{M}}. $$

On the other side, again by **point 1** we have

$$ V_{k_0}^{p_1} = Z_{k_0}^{p_1} + \sum_{p' \in \mathcal{P}} \int_{k_0}^{k_0-} a_{p'}^{p_1} h_{p'}^{p_1} (k_0 - k') V_{k'}^{p'} d\mathcal{D}_{k'}^{p'} $$
$$ 
\begin{aligned}
&= Z_{k_0}^{p_1} \\
&= 1_{p_0=p_1},
\end{aligned}
$$

which concludes on $\langle p_1, k_0 \rangle = 1_{p_0=p_1}$.

Let $(Z_k^p)_{(p,k) \in Q, k \ge k_0}$ be an arbitrary sequence, let $(p, k) \in Q$ with $k > k_0$. Then

$$ 
\begin{aligned}
V_k^p &= Z_k^p + \sum_{p'' \in \mathcal{P}} \int_{k_0}^{k-} a_{p''}^p h_{p''}^p (k-k'') V_{k''}^{p''} d\mathcal{D}_{k''}^{p''} \\
&= Z_k^p + \sum_{p'' \in \mathcal{P}} \int_{k_0}^{k-} a_{p''}^p h_{p''}^p (k-k'') \left[ \sum_{p' \in \mathcal{P}} \int_{k_0}^{k''} \left\langle \begin{pmatrix} p'', k'' \\ p', k' \end{pmatrix} \right\rangle_{\mathfrak{M}} Z_{k'}^{p'} d\mathcal{D}_{k'}^{p'} \right] d\mathcal{D}_{k''}^{p''} \\
&= Z_k^p + \sum_{p' \in \mathcal{P}} \int_{k_0}^{k-} \left[ \sum_{p'' \in \mathcal{P}} \int_{k'}^{k-} a_{p''}^p h_{p''}^p (k-k'') \left\langle \begin{pmatrix} p'', k'' \\ p', k' \end{pmatrix} \right\rangle_{\mathfrak{M}} d\mathcal{D}_{k''}^{p''} \right] Z_{k'}^{p'} d\mathcal{D}_{k'}^{p'}.
\end{aligned}
$$

By identification of the coefficients one obtains **point 3. Point 2**, aka the 1-periodicity, can be proved by looking at the sequence $\tilde{Z}$: the version of $Z$ shifted by 1. The associated $\tilde{V}$ sequence is then the sequence $V$ shifted by 1 and thus the coefficients must be 1-periodic. It can also be proved by induction for increasing values of $k - k'$ by using the formula of **point 3**. $\square$

**Proof of Proposition 4.4.** It is quite clear that the closed form satisfies the recursive property of point 3 of Lemma 4.3. Indeed, one only needs to cut the paths in two sub paths: all the first steps except the last one and the last one. Thus the closed form must be equal to the coefficients. $\square$

## 9.2. Proof of Theorem 4.5

To prove Theorem 4.5 we solve completely the recurrence relation on the chains by using coefficients and then inject back in the intensity equation these calculations. $L^1$ bounds on the cluster functions are derived with analytics properties of the coefficients derived in Lemma B.1 in Appendix B.

**Proof of Theorem 4.5.** We will prove the theorem for $t_0 = -\infty$. If $t_0 \in \mathbb{R}$ the ideas are the same and even less technical since no integrability conditions are needed. Recall that $\kappa_{p'}^p(x) = a_{p'}^p x + b_{p'}^p$. Let

$$ Z_k^p := \xi_k^p + \sum_{m \in \mathcal{M}} \int_{-\infty}^{k-} h_m^p (k-s) dS_m^m + \sum_{p' \in \mathcal{P}} b_{p'}^p \int_{-\infty}^{k-} h_{p'}^p (k-k') d\mathcal{D}_{k'}^{p'} $$

so that

$$ 
\begin{aligned}
W_k^p &= \xi_k^p + \sum_{m \in \mathcal{M}} \int_{t_0}^{k-} h_m^p (k-s) dS_m^m + \sum_{p' \in \mathcal{P}} \int_{t_0}^{k-} h_{p'}^p (k-k')(a_{p'}^p W_{k'}^{p'} + b_{p'}^p) d\mathcal{D}_{k'}^{p'} \\
&= Z_k^p + \sum_{p' \in \mathcal{P}} \int_{-\infty}^{k-} a_{p'}^p h_{p'}^p (k-k') W_{k'}^{p'} d\mathcal{D}_{k'}^{p'}
\end{aligned}
$$

One can calculate

$$ \sup_{(p,k) \in \mathcal{Q}} \mathbb{E}[|Z_k^p|] \le \sup_{p,k} \mathbb{E}[\xi_k^p] + \sum_m (\sup_p \|h_m^p\|_1) \sup_{s \in \mathbb{R}} \mathbb{E}[|\lambda_s^m|] + \sum_{p' \in \mathcal{P}} \sup_p b_{p'}^p \|h_{p'}^p\|_{1,n_{p'}}^\infty. $$

Thus, $\sup_{(p,k) \in \mathcal{Q}} \mathbb{E}[|Z_k^p|] < \infty$ by the different assumptions. Proposition B.2, in Appendix B, extends point 1 of Lemma 4.3 to random sequences and to the case $k_0 = -\infty$. Since by Theorem 3.9 we have $\sup_{(p,k) \in \mathcal{Q}} \mathbb{E}[|W_k^p|] < \infty$, combined
to sup$_{(p,k)}$$_{∈Q}$ E [|Z<sub>k</sub><sup>p</sup>|] < ∞ we can apply Proposition B.2 and we obtain that,

$$
\begin{align*}
W_k^p &= \sum_{p' \in \mathcal{P}} \int_{-\infty}^k \left\langle \begin{pmatrix} p, k \\ p', k' \end{pmatrix} \right\rangle_{\mathfrak{M}} Z_{k'}^{p'} d\mathcal{D}_{k'}^{p'} \\
&= Y_{p,k}^b(t_0) + Y_{p,k}^{\xi}(t_0) + \sum_m \int_{t_0}^{k-} Y_{p,k}^m(t_0, t) dS_t^m,
\end{align*}
$$

which proves the first result of Theorem 4.5. Then one has just to inject this expression into the $\lambda_t^m$ equation (2.2a) to get
the second result. It remains to check the $L^1$ bound.

$$
\begin{align*}
\mathfrak{h}_{m_0}^m(s,t) &= h_{m_0}^m(t-s) + \sum_{p,p' \in \mathcal{P}} \int_{s+}^{t-} \int_{s+}^{k} a_p^m h_p^m(t-k) \left\langle \begin{pmatrix} p,k \\ p',k' \end{pmatrix} \right\rangle_{\mathfrak{M}} h_{m_0}^{p'}(k'-s) d\mathcal{D}_{k'}^{p'} d\mathcal{D}_k^p \\
&= h_{m_0}^m(t-s) + \sum_{p,p' \in \mathcal{P}} \int_{s+}^{\infty} \int_{s+}^{\infty} \mathbb{1}_{k' \le k < t} a_p^m h_p^m(t-k) \left\langle \begin{pmatrix} p,k \\ p',k' \end{pmatrix} \right\rangle_{\mathfrak{M}} h_{m_0}^{p'}(k'-s) d\mathcal{D}_k^p d\mathcal{D}_{k'}^{p'}.
\end{align*}
$$

Thus,

$$
(9.1) \quad \| \mathfrak{h}_{m_0}^m (s, \cdot) \|_1 \le \| h_{m_0}^m \|_1 + \sum_{p, p' \in \mathcal{P}} \int_{s+}^{\infty} \int_{s+}^{\infty} a_p^m \| h_p^m \|_1 \mathbb{1}_{k' \le k} \left\langle \begin{pmatrix} p, k \\ p', k' \end{pmatrix} \right\rangle_{\mathfrak{M}} h_{m_0}^{p'} (k' - s) d\mathcal{D}_k^p d\mathcal{D}_{k'}^{p'}.
$$

Lemma B.1 in Appendix B states analytics results on the coefficients, in particular on sums of the coefficients. Fix $p, p' \in \mathcal{P}$,

$$
\begin{align*}
& \int_{s+}^{\infty} \int_{s+}^{\infty} \mathbb{1}_{k' \le k} \left\langle \begin{pmatrix} p, k \\ p', k' \end{pmatrix} \right\rangle_{\mathfrak{M}} h_{m_0}^{p'} (k' - s) d\mathcal{D}_k^p d\mathcal{D}_{k'}^{p'} \\
&= \sum_{k_0 \in \mathcal{D}_{p'} \cap (s, s+1]} \sum_{n \in \mathbb{N}} \int_{k_0+n}^{\infty} \left\langle \begin{pmatrix} p, k \\ p', k_0+n \end{pmatrix} \right\rangle_{\mathfrak{M}} h_{m_0}^{p'} (k_0 + n - s) d\mathcal{D}_k^p \\
&= \sum_{k_0 \in \mathcal{D}_{p'} \cap (s, s+1]} \sum_{n \in \mathbb{N}} \int_{k_0}^{\infty} \left\langle \begin{pmatrix} p, k \\ p', k_0 \end{pmatrix} \right\rangle_{\mathfrak{M}} h_{m_0}^{p'} (k_0 + n - s) d\mathcal{D}_k^p \\
&\le \|h_{m_0}^{p'}\|_{1,1} \sum_{k_0 \in \mathcal{D}_{p'} \cap (s, s+1]} \int_{k_0}^{\infty} \left\langle \begin{pmatrix} p, k \\ p', k_0 \end{pmatrix} \right\rangle_{\mathfrak{M}} d\mathcal{D}_k^p \\
&\le \|h_{m_0}^{p'}\|_{1,1} \times n_p [(\mathrm{Id} - H_W^W)^{-1}]_{pp'}
\end{align*}
$$

where we use, the fact that $\mathcal{D}_{p'} = \square_{n \in \mathbb{Z}} (\mathcal{D}_{p'} \cap (0,1] + n)$, 1-periodicity of the coefficients and point 2 of Lemma B.1.
Combining this bound with (9.1) concludes the proof. $\square$

**Proof of Proposition 4.9.** There is actually nothing to prove, the expression of $\lambda^m$ given in Theorem 4.5 is clear. $\square$

10. Proof of results of Section 5

We first prove the linear domination established in Proposition 5.2, then we prove exponential moments for linear HAR processes established in Theorem 5.3. To prove Corollary 5.5 we combine Proposition 5.2 and Theorem 5.3. finally we prove Proposition 5.6.

10.1. *Proof of Proposition* 5.2

**Proof of Proposition 5.2.** First, one can observe that $\varkappa_p^\alpha$ is defined such that

$$
|\kappa_p^\alpha(x)| \leq a_p^\alpha |x| + b_p^\alpha = \varkappa_p^\alpha(|x|), \quad \forall x \in \mathbb{R},
$$
and for $x = \emptyset$, $\kappa_p^\alpha(x) = \varkappa_p^\alpha(x) = 0$.

Thus if $x$ and $y$ are such that

$$x = \emptyset \Rightarrow y = \emptyset, \quad \text{and} \quad x \in \mathbb{R} \Rightarrow y \in \mathbb{R}, \quad |x| \le y$$

then we have $|\kappa_p^\alpha(x)| \le \varkappa_p^\alpha(y)$ for any $\alpha \in \mathcal{M} \cup \mathcal{P}$ and $p \in \mathcal{P}$.

**Point 1.** That fact that assumption (NoExpl)[$\mathfrak{M}^+$] holds is clear since it holds for $\mathfrak{M}$. It is also clear that the initial condition $\mathfrak{C}^+$ is integrable since $\mathfrak{C}$ is integrable.

Let $t_0 = T_0 < T_1 < T_2 < \dots < T_n < \dots$ be the ordered random sequence of both the points of $(S^m \cap [t_0, \infty)) \in \mathcal{M}$ of the HAR$_{\mathfrak{M}}(t_0, \mathfrak{C}, \xi)$ process and the domains of the chains: $\cup_{p \in \mathcal{P}} \mathcal{D}_p \cap [t_0, \infty)$. We will prove that the result holds for all $t \in [T_0, T_n]$, for all $C \subset [T_0, T_n]$, and all $(p, k) \in \mathcal{Q}$ with $T_0 \le k \le T_n$ by induction on $n$.

**n = 0:** It is clear that the result holds by construction of $\mathfrak{M}^+$, $\mathfrak{C}^+$ and $\zeta$.

**n → n + 1:** Let us first remark that by induction and by construction of $\mathfrak{C}^+$ we have, for all $\alpha \in \mathcal{M} \cup \mathcal{P}$ and all $(p, k) \in \mathcal{Q}$ with $k \le T_n$ (which is the same as $k < T_{n+1}$),

$$(10.1) \qquad |\kappa_p^\alpha(W_k^p)| \le \varkappa_p^\alpha((W_k^p)_{\mathfrak{M}^+}).$$

Since for $\mathfrak{M}$ there is no chain nor point in $(T_n, T_{n+1})$, we have, for $t \in (T_n, T_{n+1}]$,

$$\ell_t^m = \mu_m + \sum_{m' \in \mathcal{M}} \int_{-\infty}^{T_n} h_{m'}^m(t-s) dS_s^{m'} + \sum_{p \in \mathcal{P}} \int_{-\infty}^{T_n} h_p^m(t-k) \kappa_p^m(W_k^p) d\mathcal{D}_k^p.$$

By definition one has,

$$(\lambda_t^m)_{\mathfrak{M}^+} = \Phi^m(\mu_m) + \sum_{m' \in \mathcal{M}} \int_{-\infty}^{t-} L_m |h_{m'}^m(t-s)| d(S_s^{m'})_{\mathfrak{M}^+} \\ + \sum_{p \in \mathcal{P}} \int_{-\infty}^{t-} L_m |h_p^m(t-k)| \varkappa_p^m((W_k^p)_{\mathfrak{M}^+}) d\mathcal{D}_k^p$$

But since by induction we have every $\mathfrak{M}$-point before $T_n$ is also a $\mathfrak{M}^+$-point and that each possible $\mathfrak{M}^+$-point in $(T_n, t)$ has a non negative contribution to $(\lambda_t^m)_{\mathfrak{M}^+}$, we have,

$$(\lambda_t^m)_{\mathfrak{M}^+} \ge \Phi^m(\mu_m) + \sum_{m' \in \mathcal{M}} \int_{-\infty}^{T_n} L_m |h_{m'}^m(t-s)| dS_s^{m'} + \sum_{p \in \mathcal{P}} \int_{-\infty}^{T_n} L_m |h_p^m(t-k)| \kappa_p^m(W_k^p) d\mathcal{D}_k^p.$$

Let us check that $L_m |\ell_t^m - \mu_m| \le (\lambda_t^m)_{\mathfrak{M}^+} - \Phi^m(\mu_m)$. First,

$$(10.2) \qquad \left| \sum_{m' \in \mathcal{M}} \int_{-\infty}^{T_n} h_{m'}^m(t-s) dS_s^{m'} \right| \le \sum_{m' \in \mathcal{M}} \int_{-\infty}^{T_n} |h_{m'}^m(t-s)| dS_s^{m'}.$$

Secondly,

$$\left| \sum_{p \in \mathcal{P}} \int_{-\infty}^{T_n} h_p^m(t-k) \kappa_p^m(W_k^p) d\mathcal{D}_k^p \right| \le \sum_{p \in \mathcal{P}} \int_{-\infty}^{T_n} |h_p^m(t-k)| \kappa_p^m(W_k^p) d\mathcal{D}_k^p.$$

Combining this inequality with (10.2) and the lower bound on $(\lambda_t^m)_{\mathfrak{M}^+}$ we obtain the result. Then, since $\Phi^m$ is $L_m$ Lipschitz, we have

$$\lambda_t^m = \Phi^m(\ell_t^m) \le \Phi^m(\mu_m) + L_m |\ell_t^m - \mu_m| \le (\lambda_t^m)_{\mathfrak{M}^+}.$$

Therefore in $(T_n, T_{n+1}]$ every $\mathfrak{M}$-point is also an $\mathfrak{M}^+$-point. Finally, if there exists $p \in \mathcal{P}$ such that $(p, T_{n+1}) \in \mathcal{Q}$ we have

$$|W_{T_{n+1}}^p| = \left| \xi_{T_{n+1}}^p + \Phi^p \left( \sum_m \int_{-\infty}^{T_{n+1}-} h_m^p(T_{n+1}-s) dS_s^m + \sum_{p'} \int_{-\infty}^{T_{n+1}-} h_{p'}^p(T_{n+1}-k') \kappa_{p'}^p(W_{k'}^p) d\mathcal{D}_{k'}^{p'} \right) \right|$$
$$
\begin{align*}
&\le \zeta_{T_{n+1}}^p + \sum_m \int_{-\infty}^{T_{n+1}-} L_p |h_m^p(T_{n+1}-s)| d(S_s^m)_{\mathfrak{M}^+} \\
&\qquad + \sum_{p'} \int_{-\infty}^{T_{n+1}-} L_p |h_{p'}^p(T_{n+1}-k')| \varkappa_{p'}^p((W_{k'}^{p'})_{\mathfrak{M}^+}) d\mathcal{D}_{k'}^{p'} \\
&= (W_{T_{n+1}}^p)_{\mathfrak{M}^+},
\end{align*}
$$

which concludes the proof of **point 1**.

**Point 2.** First let us check that $\mathfrak{M}^+$ satisfies the assumptions of Theorem 3.9. The assumptions on the random drifts (FiMe)[$\zeta$] is clear since $\zeta = |\xi|$. Then an easy check shows that $H_{\mathfrak{M}^+} = H_{\mathfrak{M}}$ thus assumption (Spec1)[$\mathfrak{M}^+$] is also clear.

Finally one easily checks by induction on $n$, using the exact same ideas of the proof of **point 1**, that the following is true in the Picard iteration from the proof of Theorem 3.9,

$$
\forall n \in \mathbb{N}, \quad \left\{
\begin{array}{l}
\forall m \in \mathcal{M}, \quad \forall t \in \mathbb{R}, \quad L_m |\ell_t^{m,n} - \mu_m| \le (\lambda_t^{m,n})_{\mathfrak{M}^+} - \Phi^m(\mu_m) \\
\forall m \in \mathcal{M}, \quad \forall t \in \mathbb{R}, \quad \lambda_t^{m,n} \le (\lambda_t^{m,n})_{\mathfrak{M}^+} \\
\forall (p,k) \in \mathcal{Q}, \quad |W_k^{p,n}| \le (W_k^{p,n})_{\mathfrak{M}^+}
\end{array}
\right..
$$

Since the result is true at each step of the Picard iteration, it is true for the limit, which concludes.

<figure><img src="image_6.png" /></figure>

10.2. *Proof of Theorem 5.3*

The proof of Theorem 5.3 is based on Lemma C.1, in Appendix C, which derives functional exponential moments for
linear combinations of the random drifts and a large class of point processes with independent clusters defined in Section
4.3 and immigrants rates depending linear combinations of the random drifts.

**Proof of Theorem 5.3.** First let us remark that under assumption (Spec2)[M], we have ‖h_m^p‖_1,1 < ∞ and thus for any n ∈ N* we have, for any m, p ∈ M, P,

$$
(10.3) \qquad \|h_m^p\|_{1,1/n}^\infty \le \|h_m^p\|_{1,1}^\infty < \infty.
$$

We continue by fitting the immigrant rate to the framework of Lemma C.1. From Theorem 4.5,

$$
\mathcal{I}_t^m = \mu_m + \sum_{p \in \mathcal{P}} \int_{-\infty}^{t-} \left[ h_p^m (t-k) \times (a_p^m Y_{p,k}^b + a_p^m Y_{p,k}^\xi + b_p^m) \right] d\mathcal{D}_k^p
$$

is the immigration part of the intensity at time *t* of node *m*. We want to apply Lemma **C.1**.

From the definition of $Y_{p,k}^b$ and point 1 of Lemma B.1 (the coefficients are summable), we have

$$
\begin{align*}
\sup_{(p,k) \in \mathcal{Q}} Y_{p,k}^b &= \sup_{(p,k) \in \mathcal{Q}} \sum_{p' \in \mathcal{P}} \int_{-\infty}^k \left\langle p, k \right\rangle \mathfrak{M} \left[ \sum_{p'' \in \mathcal{P}} b_{p''}^{p'} \int_{-\infty}^{k'-} h_{p''}^{p'} (k' - k'') d\mathcal{D}_{k''}^{p''} \right] d\mathcal{D}_{k'}^{p'} \\
&\le P^2 \sup_{p,p',p'' \in \mathcal{P}} (\mathrm{Id} - H_W^W)^{-1}_{p,p'} b_{p''}^{p'} \|h_{p''}^{p'}\|_{1,n_{p''}}^{\infty}.
\end{align*}
$$

Thus it is clear that $\sup_{p,k} Y_{p,k}^b < \infty$. From assumption (Spec1)[M], we have $\|h_p^m\|_{1,n_p}^\infty < \infty$ for all $m, p \in M, P$, thus we have

$$
(10.4) \qquad i_m := \sup_{t \in \mathbb{R}} \left( \mu_m + \sum_{p \in \mathcal{P}} \int_{-\infty}^{t-} \left[ h_p^m (t-k) \times (a_p^m Y_{p,k}^b + b_p^m) \right] d\mathcal{D}_k^p \right) < \infty.
$$
Recall that $Y_{p,k}^{\xi} = \sum_{p' \in \mathcal{P}} \int_{-\infty}^{k} \langle p, k \rangle \xi_{k'}^{p'} d\mathcal{D}_{k'}^{p'}. Thus it follows that

$$ (10.5) \qquad \mathcal{I}_t^m \le i_m + \sum_{(p', k') \in \mathcal{Q}} A_{p', k'}^m(t) \xi_{k'}^{p'} $$

with $A_{p',k'}^m(t) = \sum_{p \in \mathcal{P}} \int_{-\infty}^{t-} 1_{k' \le k < t} a_p^m h_p^m(t-k) \langle p, k \rangle \mathcal{D}_{k'}^p d\mathcal{D}_k^p$.

Let $t \in \mathbb{R}$, $p' \in \mathcal{P}$ and $m \in \mathcal{M}$, applying point 1 of Lemma B.1 leads to,

$$ \begin{aligned} \int_{\mathbb{R}} A_{p', k'}^m(t) d\mathcal{D}_{k'}^{p'} &= \sum_{p \in \mathcal{P}} \int_{-\infty}^{t-} \int_{-\infty}^{k} a_p^m h_p^m(t-k) \langle p, k \rangle \mathcal{D}_{k'}^{p'} d\mathcal{D}_k^p \\ &\le \sum_{p \in \mathcal{P}} \int_{-\infty}^{t-} a_p^m h_p^m(t-k) (\mathrm{Id} - H_W^W)^{-1}_{p, p'} d\mathcal{D}_k^p \\ &\le \sum_{p \in \mathcal{P}} a_p^m \|h_p^m\|_{1, n_p}^{\infty} (\mathrm{Id} - H_W^W)^{-1}_{p, p'}. \end{aligned} $$

Finally we obtain

$$ (10.6) \qquad \sup_{t \in \mathbb{R}} \int_{\mathbb{R}} A_{p', k'}^m(t) d\mathcal{D}_{k'}^{p'} < \infty. $$

With the notation of Lemma C.1 we have

$$ A_{\infty} := \left( \sup_{t \in \mathbb{R}} \int_{\mathbb{R}} A_{p', k'}^m(t) d\mathcal{D}_{k'}^{p'} \right)_{p', m \in \mathcal{P}, \mathcal{M}} $$

Thus we proved that

$$ A_{\infty} \preceq [(\mathrm{Id} - H_W^W)^{-1}]^{\mathrm{T}} (a_p^m \|h_p^m\|_{1, n_p}^{\infty})_{p, m \in \mathcal{P}, \mathcal{M}}. $$

The intensity fits now the framework of Lemma C.1.

For exponential moments of $F \bullet S$ we can just apply Lemma C.1 with $B^m = F^m$ and $C^p = 0$. The result follows immediately since by assumption (SubG), $K_1 := \max_{p \in \mathcal{P}} \max(\epsilon_p, \delta_p) < \infty$, and one can take $K := \max(K_1, \max_m i_m)$.

For exponential moments of $F \bullet W$ we need to work on the expression of $F \bullet W$. Recall from Theorem 4.5 the expression of $W_k^p$,

$$ W_k^p = Y_{p,k}^b + Y_{p,k}^{\xi} + \sum_m \int_{-\infty}^{k-} Y_{p,k}^m(t) dS_t^m. $$

Thus we can decompose $F \bullet W$ as follows,

$$ F \bullet W = \sum_{p \in \mathcal{P}} \int F^p(k) Y_{p,k}^b d\mathcal{D}_k^p + \sum_{p, p' \in \mathcal{P}} \int \int F^p(k) \langle p, k \rangle \xi_{k'}^{p'} d\mathcal{D}_k^p d\mathcal{D}_{k'}^{p'} \\ + \sum_{m \in \mathcal{M}} \int \sum_{p, p' \in \mathcal{P}} \int \int F^p(k) \langle p, k \rangle h_m^{p'}(k' - t) d\mathcal{D}_k^p d\mathcal{D}_{k'}^{p'} dS_t^m. $$

Thus we have

$$ (10.7) \qquad F \bullet W \le K |F_{\infty}^{\mathcal{P}}|_1 + \sum_{p' \in \mathcal{P}} \int C_{k'}^{p'} \xi_{k'}^{p'} d\mathcal{D}_{k'}^{p'} + \sum_{m \in \mathcal{M}} \int B^m(t) dS_t^m $$
with $K < \infty$ depending on $\mathfrak{M}$ only and,

$$C_{k'}^{p'} = \sum_{p \in \mathcal{P}} \int F^p(k) \left\langle \begin{pmatrix} p, k \\ p', k' \end{pmatrix} \right\rangle_{\mathfrak{M}} d\mathcal{D}_k^p, \\ B^m(t) = \sum_{p, p' \in \mathcal{P}} \int \int F^p(k) \left\langle \begin{pmatrix} p, k \\ p', k' \end{pmatrix} \right\rangle_{\mathfrak{M}} h_m^{p'}(k' - t) d\mathcal{D}_{k'}^{p'} d\mathcal{D}_k^p.$$

Let $n \in \mathbb{N}^*$. With the notation of Lemma C.1 we have to bound $C_\infty$ and $B_{\infty,n}$. One has,

$$\int C_{k'}^{p'} d\mathcal{D}_{k'}^{p'} = \sum_{p \in \mathcal{P}} \int \int F^p(k) \left\langle \begin{pmatrix} p, k \\ p', k' \end{pmatrix} \right\rangle_{\mathfrak{M}} d\mathcal{D}_{k'}^{p'} d\mathcal{D}_k^p \\ \le \sum_{p \in \mathcal{P}} \int \int F^p(k) (\mathrm{Id} - H_W^W)^{-1}_{p, p'} d\mathcal{D}_k^p \\ \le ((\mathrm{Id} - H_W^W)^{-1})^\top F_\infty^P_{p'}.$$

Thus we have $C_\infty \le [(\mathrm{Id} - H_W^W)^{-1}]^\top F_\infty^P$.

For $B_{\infty,n}$ we have,

$$(B_{\infty,n})_m \le \sum_{p, p' \in \mathcal{P}} \int \int F^p(k) \left\langle \begin{pmatrix} p, k \\ p', k' \end{pmatrix} \right\rangle_{\mathfrak{M}} \|h_m^{p'}\|_{1,1/n}^\infty d\mathcal{D}_{k'}^{p'} d\mathcal{D}_k^p \\ \le \sum_{p, p' \in \mathcal{P}} \int F^p(k) (\mathrm{Id} - H_W^W)^{-1}_{p, p'} \|h_m^{p'}\|_{1,1/n}^\infty d\mathcal{D}_k^p \\ \le \sum_{p, p' \in \mathcal{P}} (F_\infty^P)_p (\mathrm{Id} - H_W^W)^{-1}_{p, p'} \|h_m^{p'}\|_{1,1/n}^\infty.$$

Which proves that $B_{\infty,n} \le R_n F_\infty^P$. Similarly, applying Lemma C.1 concludes. $\square$

## 10.3. Proof of Corollary 5.5 and Proposition 5.6

**Proof of Corollary 5.5.** Under assumption (Spec2)[$\mathfrak{M}^+$], Theorem 3.1 of [41] shows that there exists $\theta_0 > 0$ and $\theta_1 < \infty$ depending on $\mathfrak{M}$ such that if $|u|_\infty \le \theta_0$ then $|\mathcal{L}(u)|_\infty \le \theta_1$. By Jensen inequality we have $\mathcal{L}(tu) \le t\mathcal{L}(u)$ for all $0 \le t \le 1$. It follows that if $|u|_\infty \theta_0^{-1} = \beta \le 1$, we have

$$(10.8) \qquad |e^{\mathcal{L}(u)} - 1|_{\infty} \le \theta_1 e^{\theta_1 \beta}.$$

We apply Theorem 5.3 to the HAR process $\mathbf{HAR}_{\mathfrak{M}^+}(-\infty)$, which dominates, by Proposition 5.2, the HAR process $\mathbf{HAR}_{\mathfrak{M}^-}(-\infty)$. From the definition of $\mathfrak{M}^+$ it is clear that the matrices $R_n, A_\infty$ calculated in Corollary 5.5 with $\mathfrak{M}$ and in Theorem 5.3 with $\mathfrak{M}^+$ coincide. Finally one can check, using (10.8), the following bounds,

$$|\chi_n^M|_1 \le K_1 \beta n, \\ |A_\infty \chi_n^M|_1 \le K_2 \beta n, \\ |A_\infty \chi_n^M|_2^2 \le K_3 \beta^2 n^2,$$

with $K_1, K_2, K_3$ constants depending on $\mathcal{M}$, and similarly for $\chi_n^P$, which concludes the proof. $\square$

**Proof of Proposition 5.6.** The fact that $\mathrm{SpR}(\tilde{H}_\mathfrak{M}) < 1 \Rightarrow (\mathrm{Spec}1)[\mathfrak{M}]$ is a consequence of $\tilde{H}_\mathfrak{M} \succeq H_\mathfrak{M}$. Indeed, for matrix with non negative entries the spectral radius is increasing with the entries. Thus $\mathrm{SpR}(\tilde{H}_\mathfrak{M}) \ge \mathrm{SpR}(H_\mathfrak{M})$, which implies (Spec1)[$\mathfrak{M}$]. For (Spec2)[$\mathfrak{M}^+$] it is a direct consequence of (5.1), stating that

$$\mathfrak{H}_{\mathfrak{M}^+} \le H_S^S + H_W^S (\mathrm{Id} - H_W^W)^{-1} \tilde{H}_S^W,$$
and the following applied to $\tilde{H}_{\mathfrak{M}}$.

For any matrix $N = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$ with non negative entries, if $\text{SpR}(N) < 1$ then we have

$$ \text{SpR} (A + B(\text{Id} - D)^{-1}C = A + B \sum_{n \in \mathbb{N}} D^n C) \leq \text{SpR}(N). $$

Indeed, one can check that for all $i \in \mathbb{N}$ we have the following

$$ \sum_{j \geq i} \left( A + B \sum_{n \in \mathbb{N}} D^n C \right)^j \leq (\text{Id} \ 0) \left[ \sum_{j \geq i} N^j \right] \begin{pmatrix} \text{Id} \\ 0 \end{pmatrix}. $$

Thus one concludes with the following result: For any square matrix $R$ with non negative coefficients, for any norm, we have

$$ \forall 0 \le x < 1, \quad \text{SpR}(R) \le x \iff \forall \varepsilon > 0, \limsup_{n \to \infty} \frac{\| \sum_{j \ge n} R^j \|}{(x + \varepsilon)^n} = 0. $$

□

# 11. Proof of results of Section 6

## 11.1. Proof of Proposition 6.1

**Proof of Proposition 6.1.** Let $\mathfrak{M}_c$ be parameters being equal to $\mathfrak{M}$ except that $h_{p'}^p$ is replaced by $x \mapsto e^{cx} h_{p'}^p(x)$ for all $p, p' \in \mathcal{P}$ where $c > 0$ will be chosen later. Then, from the closed form of the coefficients (see Proposition 4.4) it is clear that we have

$$ (11.1) \qquad \left\langle \begin{pmatrix} p, k \\ p', k' \end{pmatrix} \right\rangle_{\mathfrak{M}_c} = e^{c(k-k')} \left\langle \begin{pmatrix} p, k \\ p', k' \end{pmatrix} \right\rangle_{\mathfrak{M}}. $$

Considering assumption (ExpTail)[$\mathfrak{M}$], we have that the function

$$ c \mapsto \|e^{cid} h_{p'}^p\|_{1, n_{p'}}^{\infty} $$

is continuous on a open set containing 0. Thus, since the spectral radius is a continuous function of the entries of the matrix, (Spec1.1)[$\mathfrak{M}$] ensures that (Spec1.1)[$\mathfrak{M}_c$] holds for $c$ small enough. Thus one can apply point 1 of Lemma B.1 and find that

$$ \int_{-\infty}^{k} \left\langle \begin{pmatrix} p, k \\ p', k' \end{pmatrix} \right\rangle_{\mathfrak{M}_c} d\mathcal{D}_{k'}^{p'} \le K < \infty $$

with $K$ (which depends on $\mathfrak{M}$ and $c$) uniform over $(p, k) \in \mathcal{Q}$. The first result follows from (11.1).

For the cluster functions, the result is a direct consequence of (ExpTail)[$\mathfrak{M}$], exponential decay of the coefficients we just proved and the expression of the clusters functions given in Definition 4.7. Indeed, for $t > s$, one has (the constants $K, c > 0$ might change from line to line),

$$ \begin{align*} h_{m'}^m(s, t) &= h_{m'}^m(t-s) + \sum_{p, p' \in \mathcal{P}} \int_{s+}^{t-} \int_{s+}^{k} a_p^m h_p^m(t-k) \left\langle \begin{pmatrix} p, k \\ p', k' \end{pmatrix} \right\rangle_{\mathfrak{M}} h_{m'}^{p'}(k'-s) d\mathcal{D}_{k'}^{p'} d\mathcal{D}_k^p \\ &\le K e^{c(t-s)} + \sum_{p, p' \in \mathcal{P}} \int_{s+}^{t-} \int_{s+}^{k} K e^{c(t-k)} K e^{c(k-k')} K e^{c(k'-s)} d\mathcal{D}_{k'}^{p'} d\mathcal{D}_k^p \\ &\le K e^{c(t-s)} + K e^{c(t-s)} \sum_{p, p' \in \mathcal{P}} \int_{s+}^{t-} \int_{s+}^{k} d\mathcal{D}_{k'}^{p'} d\mathcal{D}_k^p \\ &\le K e^{c(t-s)} + K(t-s)^2 e^{c(t-s)} \\ &\le K e^{c(t-s)}. \end{align*} $$
<figure><img src="image_7.png" /></figure>

## 11.2. Proof of Theorem 6.3

In this subsection we prove Theorem 6.3. Along this subsection we fix some parameters

$$ \mathfrak{M} = ((\mu_m)_m, (h_{\beta}^{\alpha})_{\alpha, \beta}, (\Phi^{\alpha})_{\alpha}, (\kappa_p^{\alpha})_{\alpha, p}), $$

a time $t_0 \in \mathbb{R}$ and two integrable initial conditions for $\mathfrak{M}$ at time $t_0$, $\mathfrak{C}_1$ and $\mathfrak{C}_2$. We suppose that assumptions (NoExpl) $[\mathfrak{M}]$, (Spec1.1) $[\mathfrak{M}]$, (ExpTail) $[\mathfrak{M}]$ and (Spec2) $[\mathfrak{M}^+]$ hold. In the sequel, $K$ and $c$ are positive constants depending only on $\mathfrak{M}$. Usually $K$ is intended to be large and $c$ small. They might change from line to line.

We denote by

$$ ((\lambda^{m}, \mathfrak{C}_i)_{m \in \mathcal{M}}, (S^{m}, \mathfrak{C}_i)_{m \in \mathcal{M}}, (W^p, \mathfrak{C}_i)_{p \in \mathcal{P}}) $$

the unique non exploding $\mathbf{HAR}_{\mathfrak{M}}(t_0, \mathfrak{C}_i)$ process for $i = 1, 2$ (Theorem 3.4 applies). We define the deviation process $((\Delta\lambda^m)_{m \in \mathcal{M}}, (\Delta S^m)_{m \in \mathcal{M}}, (\Delta W^p)_{p \in \mathcal{P}})$ by

* For $t \ge t_0$ let

$$ \Delta\lambda^m(t) = |\lambda_t^{m, \mathfrak{C}_1} - \lambda_t^{m, \mathfrak{C}_2}|. $$

* For $t \ge t_0$ let

$$ d\Delta S^m(t) = |dS_t^{m, \mathfrak{C}_1} - dS_t^{m, \mathfrak{C}_2}| = \pi^m(\{t\} \times (\min(\lambda_t^{m, \mathfrak{C}_1}, \lambda_t^{m, \mathfrak{C}_2}), \max(\lambda_t^{m, \mathfrak{C}_1}, \lambda_t^{m, \mathfrak{C}_2}))) $$

* For $(p, k) \in \mathcal{Q}$ with $k \ge t_0$ let

$$ \Delta W^p(k) = |W_k^{p, \mathfrak{C}_1} - W_k^{p, \mathfrak{C}_2}|. $$

Lemma D.1 in Appendix D dominates the deviation process by a linear HAR process with parameters $\mathfrak{M}^+$ and with vanishing base rates. To avoid any confusion, we denote $g_{\beta}^{\alpha} = L_{\alpha}|h_{\beta}^{\alpha}|$ the interaction functions of $\mathfrak{M}^+$ and $g_{m'}^m$ the cluster functions associated to $\mathfrak{M}^+$ defined in Definition 4.7. Lemma D.1 states the following.

There exist Poisson random measures $\mathcal{P}^m$ for $m \in \mathcal{M}$ on $[t_0, \infty) \times \mathbb{R}_+$, independent of $\mathcal{F}_{t_0-}$ and a process $((\mathcal{L}^m)_{m \in \mathcal{M}}, (\mathcal{S}^m)_{m \in \mathcal{M}}, (\mathcal{W}^p)_{p \in \mathcal{P}})$ adapted (and previsible for $\mathcal{L}$) to the filtration $\mathcal{G}_t = \sigma(\mathcal{F}_{t_0-}, \mathcal{P}_{[t_0, t] \times \mathbb{R}_+})$ such that,

* for $m \in \mathcal{M}$, $t \ge t_0$, $\Delta\lambda^m(t) \le \mathcal{L}_t^m$,

* for $m \in \mathcal{M}$, $t \ge t_0$, $d\Delta S^m(t) \le d\mathcal{S}_t^m$,

* for $(p, k) \in \mathcal{Q}$, $k \ge t_0$, $\Delta W^p(k) \le \mathcal{W}_k^p$.

And we have, for $t \ge t_0$, $C \subset [t_0, \infty)$ and $(p, k) \in \mathcal{Q}$ with $k \ge t_0$,

$$ \left\{ \begin{array}{ll} \mathcal{L}_t^m = u_0 e^{-c(t-t_0)} + \sum_{m' \in \mathcal{M}} \int_{t_0}^{t-} \mathfrak{g}_{m'}^m(s,t) d\mathcal{S}_s^{m'} & \\ \\ \mathcal{S}^m(C) = \int_{C \times \mathbb{R}_+} \mathbb{1}_{0<x \le \mathcal{L}_s^m} d\mathcal{P}^m(s,x) & \\ \\ \mathcal{W}_k^p = u_0 e^{-c(k-t_0)} + \sum_{m \in \mathcal{M}} \int_{t_0}^{k-} K e^{-c(k-s)} d\mathcal{S}_s^m & \end{array} \right. $$

where $u_0 = K|\mathfrak{C}_1, \mathfrak{C}_2|_{\delta}$ with $K, c, \delta > 0$ constants depending only on $\mathfrak{M}$.

The process $((\mathcal{L}^m)_{m \in \mathcal{M}}, (\mathcal{S}^m)_{m \in \mathcal{M}}, (\mathcal{W}^p)_{p \in \mathcal{P}})$ is quite similar to a $\mathbf{HAR}_{\mathfrak{M}^+}(t_0)$ process. The main difference is that the immigration rate of the point processes is not constant nor 1-stationary as in Theorem 4.5, it vanishes exponentially
fast. Thus one can have the following reasoning: since the immigrant rate is vanishing, at some point there will be no more cluster roots and thus no more points when all the clusters will have ended.

Let $T_e$ the extinction time of $(\mathcal{S}^m)_{m \in M}$. It is defined as the time of the last point,

$$T_e = \max_{m \in M} \sup \{t \ge t_0 \mid d\mathcal{S}_t^m = 1\}.$$

Let us prove the following result.

**Proposition 11.1.** There exists $K, c > 0$ constants depending only on $\mathfrak{M}$ such that

$$\forall t \ge 0, \mathbb{P}(T_e \le t_0 + t) \ge 1 - K \mathrm{E}[|\mathfrak{C}_1, \mathfrak{C}_2|_\delta] e^{-ct}.$$

**Proof of Proposition 11.1.** First let us work conditionally to $u_0$, so that the immigration rate is deterministic. We use the cluster representation. Denote by $\Pi^m$ the immigrant points from node $m$. We use the same notations as in Section 4.3 so that

$$\Pi^m = \{s \in [t_0, \infty) \mid (\emptyset, m, s) \text{ is the root of a cluster}\}.$$

For an immigrant point $(\emptyset, m, s)$ we denote by $\tau(m, s)$ the (temporal) length of the cluster it generates. We emphasize that the law of the length of the cluster depends on $s$ since the interacting functions $g(s, t)$ depend on both $s$ and $t$ and not just on $t-s$. We also denote by $\tau^m(s)$ a random variable with law being the length of a cluster born at time $s$ and with a root of type $m$.

Conditionally to $u_0$ we have,

$$
\begin{aligned}
\mathbb{E}[1_{T_e \le t_0 + t}] &= \mathbb{E} \left[ \prod_{m \in M} \prod_{s \in \Pi^m} 1_{\tau(m,s) \le t_0 + t - s} \right] \\
&= \mathbb{E} \left[ \prod_{m \in M} \prod_{s \in \Pi^m} \mathbb{P}(\tau^m(s) \le t_0 + t - s) \right] \\
&= \mathbb{E} \left[ \exp \left( \sum_{m \in M} \int_{t_0}^{\infty} \log \left[ \mathbb{P}(\tau^m(s) \le t_0 + t - s) \right] d\Pi_s^m \right) \right] \\
&= \exp \left( \sum_{m \in M} \int_{t_0}^{\infty} \left( \mathbb{P}(\tau^m(s) \le t_0 + t - s) - 1 \right) u_0 e^{-c(s-t_0)} ds \right) \\
&= \exp \left( - \sum_{m \in M} \int_{t_0}^{\infty} \mathbb{P}(\tau^m(s) > t_0 + t - s) u_0 e^{-c(s-t_0)} ds \right)
\end{aligned}
$$

with $u_0 e^{-c(s-t_0)}$ the immigrants rate.

Let $G_s^m$ a cluster born at time $s$ with root of type $m$, then for $t_0 + t \ge s$ we have

$$
\begin{aligned}
\mathbb{P}(\tau^m(s) > t_0 + t - s) &= \mathbb{P}(G_s^m([s + t_0 + t - s, \infty)) \ge 1) \\
&= \mathbb{P} \left( e^{\theta_2 \mathrm{Card}(G_s^m \cap [s + t_0 + t - s, \infty))} - 1 \ge e^{\theta_2} - 1 \right) \\
&\le K e^{-c(t_0 + t - s)}
\end{aligned}
$$

where in the last step we use Markov inequality and Lemma 6.2 (which applies since $(\mathrm{ExpTail})[\mathfrak{M}]$ implies $(\mathrm{ExpTail})[\mathfrak{M}^+]$). If $s > t_0 + t$ then we trivially have $\mathbb{P}(\tau^m(s) > t_0 + t - s) = 1$. It follows that

$$\mathbb{P}(T_e \le t_0 + t) \ge \exp \left( - \int_{t_0}^{t_0+t} u_0 K e^{-c(t_0+t-s)} e^{-c(s-t_0)} ds - \int_{t_0+t}^{\infty} u_0 e^{-c(s-t_0)} ds \right).$$
From this point easy calculations leads to

$$P(T_e \le t) \ge \exp(-u_0 Ke^{-ct}) \ge 1 - u_0 Ke^{-ct}.$$

It only remains to integrate with respect to $u_0 = K|\mathfrak{C}_1, \mathfrak{C}_2|_\delta$ since we were conditionally to $u_0$, which concludes the proof of Proposition 11.1. $\square$

Proposition 11.1 is exactly the first point of Theorem 6.3 for the point process part. To control the chains, we need bounds on the total number of points. Let $\mathcal{S}_\infty = \sum_{m \in M} \mathcal{S}^m(|t_0, \infty))$ the total number of points. Let us prove that moments of $\mathcal{S}_\infty$ are linked to moments of $|\mathfrak{C}_1, \mathfrak{C}_2|_\delta$. More precisely we have the following.

**Proposition 11.2.** For all $q \ge 1$, there exists $K_q$ depending on $q$ and $\mathfrak{M}$ such that

$$E[\mathcal{S}_{\infty}^q] \le K_q E[|\mathfrak{C}_1, \mathfrak{C}_2|_{\delta} + |\mathfrak{C}_1, \mathfrak{C}_2|_{\delta}^q].$$

**Proof of Proposition 11.2.** We use the same notations as in Proposition 11.1, we denote by $\Pi^m$ all the immigrant points from node $m \in M$. For $s \in \Pi^m$, the cluster generated by $(\emptyset, m, s)$ is $G_s^m$. As in the proof of Proposition 11.1 we work conditionally to $u_0$ and integrate with respect to $u_0$ at the end. Let $\mathcal{A}^m$ be $\mathcal{Poi}_s(\mathfrak{H}_{\mathfrak{M}^+}^\top)$ Galton Watson process with root of type $m \in M$. As explains in Section 4.3, for any $s \in \Pi^m$ and any $m \in M$, there exists a coupling such that $\mathcal{A}^m$ dominates stochastically $G_s^m$.

Since conditionally to the immigrants the clusters are independent we have,

$$ \begin{aligned} E[\mathcal{S}_{\infty}^q] &= E\left[ E\left[ \left( \sum_{m \in M} \sum_{s \in \Pi^m} \mathrm{Card}(G_s^m) \right)^q \middle| (\Pi^m)_m \right] \right] \\ &\le E\left[ \left( \sum_{m \in M} \sum_{s \in \Pi^m} E[\mathrm{Card}(G_s^m)^q]^{1/q} \right)^q \right] \\ &= E\left[ \left( \sum_{m \in M} \mathrm{Card}(\Pi^m) E[\mathrm{Card}(\mathcal{A}^m)^q]^{1/q} \right)^q \right] \\ &\le E[\mathrm{Card}(\Pi^q)] \max_{m \in M} E[\mathrm{Card}(\mathcal{A}^m)^q], \end{aligned} $$

where we used Minkowski to obtain the second line and $\Pi = \bigcup_{m \in M} \Pi^m$. Since conditionally to $u_0$, $\mathrm{Card}(\Pi)$ follows a law $\mathcal{Poi}_s(Mu_0/c)$ where $u_0 = K|\mathfrak{C}_1, \mathfrak{C}_2|_\delta$ we have

$$ \begin{aligned} E[\mathrm{Card}(\Pi)^q] &= E[E[\mathcal{Poi}_s(Mu_0/c)^q | u_0]] \\ &\le K_q E[|\mathfrak{C}_1, \mathfrak{C}_2|_\delta + |\mathfrak{C}_1, \mathfrak{C}_2|_\delta^q] \end{aligned} $$

where $K_q$ is a constant depending on $q$ and $KM/c$ (which depends on $\mathfrak{M}$) such that the moment of order $q$ of the law $\mathcal{Poi}_s(\alpha KM/c)$ is bounded by $K_q(\alpha + \alpha^q)$. Which concludes the proof of Proposition 11.2. $\square$

Let us now focus on chains. Since $\mathcal{W}_k^p$ is a function of the point processes, we can use the results of Propositions 11.1 and 11.2 to obtain the following result.

**Proposition 11.3.** There exists $c, K > 0$ constants depending only on $\mathfrak{M}$ such that for all $t \ge 0$ we have

$$P\left(\sum_{p \in \mathcal{P}} \int_{t_0+t}^{\infty} \mathcal{W}_k^p d\mathcal{D}_k^p \le e^{-ct}\right) \ge 1 - K E\left[|\mathfrak{C}_1, \mathfrak{C}_2|_{\delta} e^{-ct}\right].$$

**Proof of Proposition 11.3.** Recall that

$$\mathcal{W}_k^p = u_0 e^{-c(k-t_0)} + \sum_{m \in M} \int_{t_0}^{k-} Ke^{-c(k-s)} d\mathcal{S}_s^m.$$
Thus

$$ \sum_{p \in \mathcal{P}} \int_{t_0+t}^{\infty} \mathcal{W}_k^p d\mathcal{D}_k^p \leq Ku_0 e^{-ct} + \sum_{m \in \mathcal{M}} \int_{t_0}^{\infty} Ke^{-c(t_0+t-s)} d\mathcal{S}_s^m. $$

And thus, if $\mathcal{S}_\infty$ is the total number of points, $T_e$ the extinction time then one has

$$ \sum_{p \in \mathcal{P}} \int_{t_0+t}^{\infty} \mathcal{W}_k^p d\mathcal{D}_k^p \leq Ku_0 e^{-ct} + K\mathcal{S}_\infty e^{-c(t_0+t-T_e)}. $$

From Proposition 11.1 we have

$$ \mathbb{P}(T_e > t_0 + t/4) \leq K'\mathbb{E}[|\mathfrak{C}_1, \mathfrak{C}_2|_\delta] e^{-c't}. $$

From Proposition 11.2 and Markov inequality

$$ \mathbb{P} \left( \delta_{\infty} > e^{ct/4} / 2K \right) \leq K' \mathbb{E} \left[ |\mathfrak{C}_1, \mathfrak{C}_2|_{\delta} \right] e^{-c't}. $$

Finally, Markov inequality also gives

$$ \mathbb{P} \left( u_0 > e^{ct/2} / 2K \right) \leq K' \mathbb{E} \left[ |\mathfrak{C}_1, \mathfrak{C}_2|_{\delta} \right] e^{-c't}. $$

Combining previous inequalities gives

$$ \mathbb{P} \left( Ku_0 e^{-ct} + K \mathcal{S}_\infty e^{-c(t_0+t-T_e)} > e^{-ct/2} \right) \leq K' \mathbb{E} \left[ |\mathfrak{C}_1, \mathfrak{C}_2|_\delta \right] e^{-c't}, $$

and thus,

$$ \mathbb{P} \left( \sum_{p \in \mathcal{P}} \int_{t_0+t}^{\infty} \mathcal{W}_k^p d\mathcal{D}_k^p > e^{-c/2t} \right) \leq K' \mathbb{E} \left[ |\mathfrak{C}_1, \mathfrak{C}_2|_{\delta} \right] e^{-c't}. $$

Which concludes the proof of Proposition 11.3 up to reducing either $c$ or $c'$ so that $c/2 = c'$. $\square$

The proof of Theorem 6.3 follows easily from this point.

The first point of Theorem 6.3 is exactly Propositions 11.1 and 11.3. Let us focus on the almost sure convergences. We have

$$ \sum_{m \in \mathcal{M}} \mathrm{Card} \left( (S^{m, \mathfrak{C}_1} \triangle S^{m, \mathfrak{C}_2}) \cap [t+t_0, \infty) \right) \leq \mathcal{S}_\infty \mathbb{1}_{T_e \leq t_0+t} $$

where $\mathcal{S}_\infty$ is the total number of points of $\mathcal{S}$ and $T_e$ the extinction time. Since from Proposition 11.2 the expectation of $\mathcal{S}_\infty$ is finite, it follows that $\mathcal{S}_\infty < \infty$ almost surely and thus $T_e < \infty$ almost surely which concludes the almost sure convergence. For the chains, from the proof of Proposition 11.3 we have

$$ \sum_{p \in \mathcal{P}} \int_{t_0+t}^{\infty} |W_k^{p, \mathfrak{C}_1} - W_k^{p, \mathfrak{C}_2}| d\mathcal{D}_k^p \leq u_0 K e^{-ct} + K \mathcal{S}_\infty e^{-c(t_0+t-T_e)}. $$

Finiteness of $T_e$, $\mathcal{S}_\infty$ and $u_0$ leads to the almost sure convergence. It remains the $L^q$ convergences. They are simple consequences of the following statement.

$$ (11.2) \quad \forall 1 \leq q < q_0, \forall x, z \geq 0, \quad \left( \mathbb{P}(Y > x) \leq z \implies \mathbb{E}[Y^q] \leq x^q + z^{1-q/q_0} \mathbb{E}[Y^{q_0}]^{q/q_0} \right) $$

where $q_0 > 1$ and $Y$ is non negative random variable. We apply (11.2) to

$$ Y = Y_t = \sum_{m \in \mathcal{M}} \mathrm{Card} \left( (S^{m, \mathfrak{C}_1} \triangle S^{m, \mathfrak{C}_2}) \cap [t+t_0, \infty) \right) $$
with $x = x_t = e^{-ct}$ and $z = z_t = K\mathbb{E}[|\mathfrak{C}_1, \mathfrak{C}_2|_\delta]e^{-ct}$. Since by assumption $\mathbb{E}[|\mathfrak{C}_1, \mathfrak{C}_2|_\delta^{q_0}] < \infty$ it is clear by Proposition 11.2 that $Y_t \le \mathcal{S}_\infty \in L^{q_0}$ uniformly in $t$, which gives the result. Similarly, for the chains we apply (11.2), with the same $x_t$ and $z_t$, to

$$Y_t = \sum_{p \in \mathcal{P}} \int_{t_0+t}^{\infty} |W_k^{p, \mathcal{C}_1} - W_k^{p, \mathcal{C}_2}| d\mathcal{D}_k^p \le u_0 + K \mathcal{S}_\infty \in L^{q_0},$$

where the bounds in $L^{q_0}$, uniform in $t$, are from Proposition 11.2. Which concludes on the $L^q$ convergences and finishes the proof.

### 11.3. Proof of Corollary 6.4 and Propositions 6.6 and 6.7

**Proof of Corollary 6.4.** To prove Corollary 6.4 we apply Theorem 6.3 with the initial conditions $\mathfrak{C}^{-\infty}$ and $\mathfrak{C}^{\mathcal{D}}$. The first one is given by the stationary HAR process on $(-\infty, t_0)$ and the second one is the empty one. From this choice, it is clear that the processes of interest are, respectively, the stationary process and the process started at time $t_0$. Let $\delta > 0$. We need to prove that $|\mathfrak{C}^{-\infty}, \mathfrak{C}^{\mathcal{D}}|_\delta$ has a finite mean and, under (SubG), moments of any order. We have

$$|\mathfrak{C}^{-\infty}, \mathfrak{C}^{\mathcal{D}}|_{\delta} = \max \left( \left[ \max_{m \in \mathcal{M}} \int_{-\infty}^{t_0^{-}} e^{-\delta(t_0-s)} dS_s^{m, -\infty} \right], \left[ \max_{p \in \mathcal{P}} \int_{-\infty}^{t_0^{-}} e^{-\delta(t_0-k)} (1 + |W_k^{p, -\infty}|) d\mathcal{D}_k^p \right] \right).$$

Thus from Theorem 3.9 it is clear that there is a finite expectation since $\sup_{m \in \mathcal{M}}, t \in \mathbb{R} \mathbb{E}[\lambda_t^{m, -\infty}] < \infty$ and $\sup_{(p,k) \in \mathcal{Q}} \mathbb{E}[|W_k^{p, -\infty}|] < \infty$.

If (SubG) holds then Corollary 5.5 applies and $|\mathfrak{C}^{-\infty}, \mathfrak{C}^{\mathcal{D}}|_\delta$ has exponential moments (thus moments of any order) regardless of $\delta > 0$. Corollary 6.4 follows directly from Theorem 6.3. $\square$

**Proof of Proposition 6.6.** The process $X^{(n, \ell, \phi)}|_{[jn+\phi, \infty)}$ only depends on processes $\text{HAR}_{\mathfrak{M}}(in + \phi - \ell)$ with $i \ge j$.

Thus it is clear that $X^{(n, \ell, \phi)}|_{[jn+\phi, \infty)}$ is independent of $\mathcal{F}_s$ for any $s < jn + \phi - \ell$, leading to the result.

For the second result, let

$$Y_j = X^{(n, \ell, \phi)}|_{[jn+\phi, (j+1)n+\phi+(n-\ell))}.$$

It is clear that $Y_j$ is measurable for $\mathcal{F}_{((j+1)n+\phi+(n-\ell))}$, and, by the first part, independent of $\mathcal{F}_{(jn+\phi-\ell)}$. Since

$$jn + \phi - \ell = ((j-2)+1)n + \phi + (n-\ell),$$

it is clear that $Y_j$ is independent of $Y_{j-2}$ and any $(Y_i)_{i \le j-2}$. Since $j$ is arbitrary it concludes. $\square$

**Proof of Proposition 6.7.** This result is straightforward from Corollary 6.4 since we are comparing the processes $\text{HAR}_{\mathfrak{M}}(-\infty)$ and $\text{HAR}_{\mathfrak{M}}(t_0)$ with $t_0 = jn + \phi - \ell$ on the interval $[t_0 + \ell, t_0 + \ell + n]$. $\square$

## 12. Proofs of results of Section 7

We recall the in Section 7 the observed process is the $\text{HAR}_{\mathfrak{M}^*}(-\infty)$ process from Theorem 3.9 and that assumptions (Spec1)[$\mathfrak{M}^*$], (Spec2)[$\mathfrak{M}^*, +$], (ExpTail)[$\mathfrak{M}^*$], (FiMe) and (SubG) hold. In the sequel $C^*$ usually stands for a large constant depending on $\mathfrak{M}^*$, while $c^*$ stands for a small constant depending on $\mathfrak{M}^*$. They might change from line to line.

### 12.1. Proof of Theorem 7.9 and Proposition 7.11

**Proof of Theorem 7.9.** The proof of Theorem 7.9 is based on Lemmas E.2 and E.3 given in Appendix E. These lemmas generalize Bernstein inequality to HAR processes.

Let $\varphi \in \Phi_M$, from Lemma E.2 we have that for any $x > 0$ and $q > 0$

$$|b_{\varphi} - \tilde{b}_{\varphi}| = |J_{\varphi, M}| \le B_{\varphi} \left( \sqrt{\frac{3x(x + C^*(q+1)T \log(T))}{3 - e}} + \frac{x}{3} \right)$$
with $B_\varphi = 1$ and probability at least

$$1 - 4e^{-x} \frac{\log(1 + C^*(q+1)T \log(T)/x)}{\log(3/2)} - \frac{C^*}{T^q}.$$

Let us apply this result with $x = (q+1) \log(T) > 0$ since $T \ge 2$. Then the bound becomes

$$|b_{\varphi} - \tilde{b}_{\varphi}| \le C^*(q+1) \log(T) \sqrt{T}$$

and the probability is lower bounded by $1 - C^*/T^q$ since

$$4e^{-x} \frac{\log(1 + C^*(q+1)T \log(T)/x)}{\log(3/2)} \le \frac{1}{T^q} \times \left( \frac{4 \log(1 + C^*T)}{T \log(3/2)} \right) \\ \le \frac{C^*}{T^q}.$$

For the other cases, $\varphi \in \Phi^M$, $\varphi \in \Phi_P$ or $\varphi \in \Phi^P$ the proof follows the same argument: applying Lemma E.2 or Lemma E.3 with $x = (q+1) \log(T)$. $\square$

**Proof of Proposition 7.11.** Let $x$ and $J$ such that $|J| \le s$ and $|x_{J^c}|_1 \le c|x_J|_1$. Then we have

$$x \cdot Vx = x^\top Ux + x^\top (V - U)x \\ \ge \eta \|x_J\|^2 - \varepsilon |x|_1^2 \\ \ge \eta \|x_J\|^2 - \varepsilon (c+1)^2 |x_J|_1^2 \\ \ge \eta \|x_J\|^2 - \varepsilon (c+1)^2 s \|x_J\|^2$$

where we use Cauchy-Schwarz inequality to obtain the last line. This concludes the proof. $\square$

## 12.2. Proof of Lemma 7.12

Let $n \ge 2A + 1$ an integer, and $\tau \in \mathbb{N}^*$ such that

$$T = 2n\tau + r$$

with $0 \le r < 2n$. Let $\mathbb{X}^{(n, \ell=n-A, \phi=-A)}$ the coupling from Definition 6.5. For readability we drop the indexes and denote $\mathbb{X} = \mathbb{X}^{(n, n-A, -A)}$. Since $n \ge 2A + 1$ we have $\ell \ge n/2$.

In this proof we will need to bound the HAR process $X$ over all possible sliding windows of size $A$ in the interval $[-A, T)$, to do so, we define $Y_T$ as follows. Let for $m \in \mathcal{M}$,

$$Y^m(T) = \max_{\substack{\text{intervals } I \subset [-A, T) \\ \text{such that } |I| \le A}} S^m(I)$$

and similarly, for $p \in \mathcal{P}$ and $\alpha \in \mathcal{M} \cup \mathcal{P}$,

$$Y_p^\alpha(T) = \max_{\substack{\text{intervals } I \subset [-A, T) \\ \text{such that } |I| \le A}} \int_I |\kappa_p^\alpha(W_k^p)| d\mathcal{D}_k^p.$$

Finally, let $Y_T = \max(\max_m Y^m(T), \max_{\alpha, p} Y_p^\alpha(T))$. Proposition E.1, in Appendix E, bounds $Y_T$ as follows,

$$(12.1) \qquad \forall x \ge 0, \quad \mathbb{P}(Y_T \le x) \ge 1 - C^*T e^{-c^*x}.$$

It also bounds $\mathbb{Y}_T$, defined exactly like $Y_T$ but with $\mathbb{X}$ instead of $X$, in the same way.
Let us now start the proof. Recall that

$$G_{\varphi\varphi'} = \sum_{m \in \mathcal{M}} \int_{0}^{T^{-}} \Psi^{m}(\varphi, t, X_{t-A,t}) \Psi^{m}(\varphi', t, X_{t-A,t}) dt + \sum_{p \in \mathcal{P}} \int_{0}^{T^{-}} \Psi^{p}(\varphi, k, X_{k-A,k}) \Psi^{p}(\varphi', k, X_{k-A,k}) d\mathcal{D}_{k}^{p}.$$

Define

• for $t_0 < t_1$,

$$(12.2) \quad (G_{t_0}^{t_1})_{\varphi\varphi'} := \sum_{m \in \mathcal{M}} \int_{t_0}^{t_1^{-}} \Psi^m(\varphi, t, X) \Psi^m(\varphi', t, X) dt + \sum_{p \in \mathcal{P}} \int_{t_0}^{t_1^{-}} \Psi^p(\varphi, k, X) \Psi^p(\varphi', k, X) d\mathcal{D}_k^p$$

and

$$(\mathbb{G}_{t_0}^{t_1})_{\varphi\varphi'} := \sum_{m \in \mathcal{M}} \int_{t_0}^{t_1^{-}} \Psi^m(\varphi, t, \mathbb{X}) \Psi^m(\varphi', t, \mathbb{X}) dt + \sum_{p \in \mathcal{P}} \int_{t_0}^{t_1^{-}} \Psi^p(\varphi, k, \mathbb{X}) \Psi^p(\varphi', k, \mathbb{X}) d\mathcal{D}_k^p$$

• For integers $1 \le j \le \tau$,

$$(\mathcal{G}_j^{\text{even}})_{\varphi\varphi'} := (\mathbb{G}_{2(j-1)n}^{(2j-1)n})_{\varphi\varphi'} - \mathbb{E} \left[ \left( \mathbb{G}_{2(j-1)n}^{(2j-1)n} \right)_{\varphi\varphi'} \right]$$

and

$$(\mathcal{G}_j^{\text{odd}})_{\varphi\varphi'} := (\mathbb{G}_{(2j-1)n}^{2jn})_{\varphi\varphi'} - \mathbb{E} \left[ \left( \mathbb{G}_{(2j-1)n}^{2jn} \right)_{\varphi\varphi'} \right].$$

Considering the support assumption, for any $t_0 < t_1$, the matrix $\mathbb{G}_{t_0}^{t_1}$ is measurable with respect to $\mathbb{X}_{t_0-A,t_1}$. The choice of the coupling $\mathbb{X} = \mathbb{X}^{n,n-A,-A}$ is tuned so that, by Proposition 6.6, the random variables $\mathcal{G}_j^{\text{even}}$, $j = 1, \dots, \tau$ are iid and similarly $\mathcal{G}_j^{\text{odd}}$, $j = 1, \dots, \tau$ are also iid.

Our goal is to control $|G_{\varphi\varphi'} - \mathbb{E}[G_{\varphi\varphi'}]|$. Thus we introduce our coupling to reveal independence to be able to apply a concentration result. First let us write

$$\begin{align} |G_{\varphi\varphi'} - \mathbb{E}[G_{\varphi\varphi'}]| &\le |(G_0^{2n\tau})_{\varphi\varphi'} - (\mathbb{G}_0^{2n\tau})_{\varphi\varphi'}| \nonumber \\ &\quad + |(\mathbb{G}_0^{2n\tau})_{\varphi\varphi'} - \mathbb{E}[(\mathbb{G}_0^{2n\tau})_{\varphi\varphi'}]| \nonumber \\ &\quad + |\mathbb{E}[(G_0^{2n\tau})_{\varphi\varphi'} - (\mathbb{G}_0^{2n\tau})_{\varphi\varphi'}]| \nonumber \\ &\quad + |(G_{2n\tau}^T)_{\varphi\varphi'} - \mathbb{E}[(G_{2n\tau}^T)_{\varphi\varphi'}]|. \tag{12.3} \end{align}$$

The ideas are the following: the first term is small since the coupling $\mathbb{X}$ is close to the stationary process $X$, the second term is small since it is equal to

$$\sum_{j=1}^{\tau} (\mathcal{G}_j^{\text{even}})_{\varphi\varphi'} + \sum_{j=1}^{\tau} (\mathcal{G}_j^{\text{odd}})_{\varphi\varphi'}$$

where both $(\mathcal{G}_j^{\text{even}})_{\varphi\varphi'}$, $j = 1, \dots, \tau$ and $(\mathcal{G}_j^{\text{odd}})_{\varphi\varphi'}$, $j = 1, \dots, \tau$ are iid centered. The third term is small since $\mathbb{X}$ is also close to $X$ in $L^2$. Finally, the last term is also relatively small since it is a centered random variable with typical size $r < n$ where the others have typical size $T$.

Proving Lemma 7.12 consists in controlling each of the four terms. The first term is controlled by the following result.

**Proposition 12.1.** For all $x \ge 0$ we have

$$\mathbb{P} \left( \forall \varphi, \varphi', \left| (G_0^{2n\tau})_{\varphi\varphi'} - (\mathbb{G}_0^{2n\tau})_{\varphi\varphi'} \right| \le T x e^{-c^* n} \right) \ge 1 - C^* T e^{-c^* x} - C^* \tau e^{-c^* n}.$$

In particular, we have

$$\mathbb{P} \left( \forall \varphi, \varphi', \left| (G_0^{2n\tau})_{\varphi\varphi'} - (\mathbb{G}_0^{2n\tau})_{\varphi\varphi'} \right| \le C^* T e^{-c^* n} \right) \ge 1 - C^* \tau e^{-c^* n}.$$
**Proof of Lemma 12.1.** Recall that $\ell = n - A$ and that $n \ge 2A + 1$ so that $\ell \ge n/2$. Remark that for $\alpha \in M \cup P$ we have

$$ (12.4) \qquad |\Psi^\alpha(\varphi, s, X)\Psi^\alpha(\varphi', s, X) - \Psi^\alpha(\varphi, s, \mathbb{X})\Psi^\alpha(\varphi', s, \mathbb{X})| \le |\Psi^\alpha(\varphi, s, X)| \times |\Psi^\alpha(\varphi', s, X) - \Psi^\alpha(\varphi', s, \mathbb{X})| \\ \qquad \qquad + |\Psi^\alpha(\varphi', s, \mathbb{X})| \times |\Psi^\alpha(\varphi, s, X) - \Psi^\alpha(\varphi, s, \mathbb{X})|. $$

Let $\varphi \in \Phi_m$ with $m \in M$, and $s \in [0, T)$. Then $\varphi_m$, the only non zero coordinate of $\varphi$ is just a base rate for node $m$, ie just a constant. Then we have $\Psi^m(\varphi, s, X) = \varphi_m$. Since $\|\Phi\|_\infty \le 1$ we have

$$ |\Psi^m(\varphi, s, X)| = |\varphi_m| \le 1. $$

Similarly, if $\varphi \in \Phi_m^p$ with $m \in M$ and $p \in P$, and $k \in [0, T) \cap D_p$, then $\varphi$ is an interaction function of node $m$ on chain $p$. Thus we have

$$ \Psi^p(\varphi, k, X) = \int_{k-A}^{k-A} \varphi_m^p(k-s) dS_m^p. $$

Since $\|\Phi\|_\infty \le 1$ we have

$$ |\Psi^p(\varphi, s, X)| \le Y_T. $$

The same reasoning holds for any $\varphi \in \Phi$, any $\alpha \in M \cup P$ and any $s \in [0, T)$, and we have

$$ (12.5) \qquad |\Psi^\alpha(\varphi, s, X)| \le 1 + Y_T. $$

Similarly,

$$ (12.6) \qquad |\Psi^\alpha(\varphi', s, \mathbb{X})| \le 1 + \mathbb{Y}_T. $$

Remark now that, if for example $\varphi \in \Phi_m^\alpha$, then

$$ |\Psi^\alpha(\varphi, s, X) - \Psi^\alpha(\varphi, s, \mathbb{X})| \le \left| \int_{s-A}^{s-A} (\kappa_p^\alpha(W_k^p) - \kappa_p^\alpha(W_k^p)) d\mathcal{D}_k^p \right| $$

and similarly there is a bound by the number of non common points between $X$ and $\mathbb{X}$ if $\varphi \in \Phi_m^\alpha$. For $j = 0, \dots, 2\tau + 1$ define $Z_j$ by

$$ Z_j = \sum_{m \in M} \mathrm{Card} \left( S^m \triangleq \mathbb{S}^m \cap [jn - A, jn + n - A) \right) + \sum_{p \in P} \int_{jn - A}^{(jn + n - A) -} |W_k^p - W_k^p| d\mathcal{D}_k^p. $$

Thus, in all the cases, the following inequality holds for all $s \in [0, T)$,

$$ (12.7) \qquad |\Psi^\alpha(\varphi, s, X)\Psi^\alpha(\varphi', s, X) - \Psi^\alpha(\varphi, s, \mathbb{X})\Psi^\alpha(\varphi', s, \mathbb{X})| \le C^*(1 + Y_T + \mathbb{Y}_T) \times \left( \max_{j=0, \dots, 2\tau+1} Z_j \right) $$

In (12.1), if $C^*$ is large enough ($C^* \ge 1/T$) then the bound trivially holds for $x \le 0$. So let us apply (12.1) to $Y_T$ and $\mathbb{Y}_T$ with $x - 1/2$ where $x \ge 0$. We also apply Proposition 6.7 to each of the $Z_j$ to obtain

$$ \mathbb{P} \left( \sup_{s \in [0, T)} |\Psi^\alpha(\varphi, s, X)\Psi^\alpha(\varphi', s, X) - \Psi^\alpha(\varphi, s, \mathbb{X})\Psi^\alpha(\varphi', s, \mathbb{X})| \le C^* x e^{-c^* \ell} \right) \ge 1 - C^* T e^{-c^* x} - C^* \tau e^{-c^* \ell}. $$

Since $\ell \ge n/2$, up to reducing $c^*$ we have

$$ \mathbb{P} \left( \sup_{s \in [0, T)} |\Psi^\alpha(\varphi, s, X)\Psi^\alpha(\varphi', s, X) - \Psi^\alpha(\varphi, s, \mathbb{X})\Psi^\alpha(\varphi', s, \mathbb{X})| \le C^* x e^{-c^* n} \right) \ge 1 - C^* T e^{-c^* x} - C^* \tau e^{-c^* n}. $$

Finally, since

$$ |(G_0^{2n\tau})_{\varphi\varphi'} - (G_0^{2n\tau})_{\varphi\varphi'}| \le (1 + \max_p n_p) \times T \times \sup_{s \in [0, T)} |\Psi^\alpha(\varphi, s, X)\Psi^\alpha(\varphi', s, X) - \Psi^\alpha(\varphi, s, \mathbb{X})\Psi^\alpha(\varphi', s, \mathbb{X})|. $$
we obtain the following

$$P\left(\left|\left(G_0^{2n\tau}\right)_{\varphi\varphi'} - \left(\mathbb{G}_0^{2n\tau}\right)_{\varphi\varphi'}\right| \le C^*Tx e^{-c^*n}\right) \ge 1 - C^*Te^{-c^*x} - C^*\tau e^{-c^*n}.$$

The proof of the first result is complete up to the change of variable $\tilde{x} = C^*x$. For the second inequality, take $x = n$ and up to decreasing $c^*$ in the LHS, we have that $ne^{-c^*n} \le C^*e^{-c^*n}$. Remark also that since $T = 2n\tau + r$, it is clear that $Te^{-c^*n} \le C^*\tau e^{-c^*n}$ up to decreasing $c^*$ in the RHS, which concludes the proof. $\square$

The next proposition deals with the second term and relies on Fuk-Nagaev concentration inequality taken from [44].

**Proposition 12.2.** For all $q > 2$ there exists $C_q^*$ and $c_q^*$ constants that depend on both $\mathfrak{M}^*$ and $q$ such that for all $x > 0$,

$$P\left(\left|\left(\mathbb{G}_0^{2n\tau}\right)_{\varphi\varphi'} - \mathbb{E}\left[\left(\mathbb{G}_0^{2n\tau}\right)_{\varphi\varphi'}\right]\right| \ge x\right) \le \frac{C_q^*\tau n^q}{x^q} + 4\exp\left(-\frac{c_q^*x^2}{\tau n^2}\right).$$

**Proof of Proposition 12.2.** With our notations,

$$\left(\mathbb{G}_0^{2n\tau}\right)_{\varphi\varphi'} - \mathbb{E}\left[\left(\mathbb{G}_0^{2n\tau}\right)_{\varphi\varphi'}\right] = \sum_{j=1}^{\tau} (\mathcal{G}_j^{\text{even}})_{\varphi\varphi'} + \sum_{j=1}^{\tau} (\mathcal{G}_j^{\text{odd}})_{\varphi\varphi'}$$

with $((\mathcal{G}_j^{\text{even}})_{\varphi\varphi'})_j$ iid random variables, as well as $((\mathcal{G}_j^{\text{odd}})_{\varphi\varphi'})_j$ and $(\mathcal{G}_j^{\text{even}})_{\varphi\varphi'} \sim (\mathcal{G}_{j'}^{\text{odd}})_{\varphi\varphi'}$.

Thus it is sufficient to look at

$$P\left(\left|\sum_{j=1}^{\tau} (\mathcal{G}_j^{\text{even}})_{\varphi\varphi'}\right| \ge x/2\right).$$

To bound this probability we want to apply Fuk Nagaev inequality (Corollary 1.8 of [44]) given in Lemma 12.3 below.

**Lemma 12.3 (Fuk-Nagaev).** Let $X_1, \dots, X_n$ be centered and independent random variables. Let $S_n = \sum_{k=1}^n X_k$. Let also $M(q, n) = \sum_{k=1}^n \mathbb{E}[|X_k|^q]$ for $q \ge 2$ and $\sigma_n^2 = \sum_{k=1}^n \mathbb{E}[X_k^2] = M(2, n)$. Then, for all $q > 2$ and $x > 0$ we have

$$P(|S_n| \ge x) \le \frac{(1+2/q)^q M(q,n)}{x^q} + 2 \exp \left( - \frac{2x^2}{(q+2)^2 e^q \sigma_n^2} \right).$$

To apply Lemma 12.3 we need to bound the moments of order $q$ for $q > 2$ and moments of order 2. Let $\tilde{q} \ge 1$. Since

$$\left(\mathcal{G}_1^{\text{even}}\right)_{\varphi\varphi'} = \sum_{k=0}^{n-1} \left(\mathbb{G}_k^{k+1}\right)_{\varphi\varphi'} - \mathbb{E}\left[\left(\mathbb{G}_k^{k+1}\right)_{\varphi\varphi'}\right]$$

we have

$$\| (\mathcal{G}_1^{\text{even}})_{\varphi\varphi'} \|_{\tilde{q}} \le n \|(\mathbb{G}_0^1)_{\varphi\varphi'}\|_{\tilde{q}} + n \|(\mathbb{G}_0^1)_{\varphi\varphi'}\|_1.$$

But since

$$\left|\left(\mathbb{G}_0^1\right)_{\varphi\varphi'}\right| \le C^* \left(1 + \sum_m S^m([-A, 1]) + \sum_p \int_{-A}^1 |W_k^p| d\mathcal{D}_k^p\right)^2$$

it is clear that $(\mathbb{G}_0^1)_{\varphi\varphi'}$ has moments of any order $\tilde{q} \ge 1$ bounded by a constant depending on both $\tilde{q}$ and $\mathfrak{M}^*$ since its square root has exponential moments by Corollary 5.5. Thus we can apply Lemma 12.3 with

$$M(q, \tau) \le C_q^* \tau n^q,$$

and

$$\sigma_n^2 = M(2, \tau) \le C^* \tau n^2,$$

which concludes the proof. $\square$