The next result controls the third term. As explain before, it is based on the fact the coupling is closed to the stationary process in $L^2$.

**Proposition 12.4.** There exist constants $C^*$ and $c^*$ such that

$$|\mathbb{E}[(G_0^{2n\tau})_{\varphi\varphi'} - (G_0^{2n\tau})_{\varphi\varphi'}]| \leq C^*Te^{-c^*n}.$$

**Proof of Proposition 12.4.** It is sufficient to prove that

$$|\mathbb{E}[(G_0^1)_{\varphi\varphi'} - (G_0^1)_{\varphi\varphi'}]| \leq C^*e^{-c^*n}.$$

As in the proof of Proposition 12.1, for any $s \in [0, 1)$, we have

$$|\Psi^{\alpha}(\varphi, s, X)\Psi^{\alpha}(\varphi', s, X) - \Psi^{\alpha}(\varphi, s, \mathbb{X})\Psi^{\alpha}(\varphi', s, \mathbb{X})| \leq |\Psi^{\alpha}(\varphi, s, X)| \times |\Psi^{\alpha}(\varphi', s, X) - \Psi^{\alpha}(\varphi', s, \mathbb{X})| \\ + |\Psi^{\alpha}(\varphi', s, \mathbb{X})| \times |\Psi^{\alpha}(\varphi, s, X) - \Psi^{\alpha}(\varphi, s, \mathbb{X})| \\ \leq C^*(1 + Y_1 + \mathbb{Y}_1) \left( \sum_m |(S^m \Delta \mathbb{S}^m)([-A, 1])| + \sum_p \int_{-A}^{1-} |W_k^p - \mathbb{W}_k^p| d\mathcal{D}_k^p \right).$$

Thus we have

$$|(G_0^1)_{\varphi\varphi'} - (G_0^1)_{\varphi\varphi'}| \leq C^*(1 + Y_1 + \mathbb{Y}_1) \left( \sum_m (S^m \Delta \mathbb{S}^m)([-A, 1]) + \sum_p \int_{-A}^{1-} |W_k^p - \mathbb{W}_k^p| d\mathcal{D}_k^p \right).$$

Clearly $(Y_1 + \mathbb{Y}_1) \in L^2$ since it even has exponential moments: indeed $X|_{[-A, 1)}$ and $\mathbb{X}|_{[-A, 1)}$ have exponential moments. Moreover, since Corollary 6.4 applies and since $\ell \geq n/2$, we have, up to lowering $c^*$,

$$\mathbb{E} \left[ \left( \sum_m (S^m \Delta \mathbb{S}^m)([-A, 1]) + \sum_p \int_{-A}^{1-} |W_k^p - \mathbb{W}_k^p| d\mathcal{D}_k^p \right)^2 \right] \leq C^* e^{-c^*\ell} \leq C^* e^{-c^*n}.$$

Cauchy Schwarz inequality concludes.

<figure><img src="image_8.png" /></figure>

Finally we finish with the fourth term.

**Proposition 12.5.** For all $q \geq 1$, there exists a constant $C_q^*$ such that for all $x > 0$,

$$\mathbb{P}(|(G_{2n\tau}^T)_{\varphi\varphi'} - \mathbb{E}[(G_{2n\tau}^T)_{\varphi\varphi'}]| \geq x) \leq \frac{C_q^*r^q}{x^q} \leq \frac{C_q^*n^q}{x^q}.$$

**Proof of Proposition 12.5.** As in the proof of Proposition 12.4, it is clear that $(G_{2n\tau}^T)_{\varphi\varphi'}$ has moments of any order bounded by $C_q^*(T - 2n\tau)^q = C_q^*r^q \leq C_q^*n^q$. Thus we have that

$$\mathbb{E} \left[ \left| (G_{2n\tau}^T)_{\varphi\varphi'} - \mathbb{E} \left[ (G_{2n\tau}^T)_{\varphi\varphi'} \right] \right|^q \right] \leq C_q^* n^q.$$

Markov inequality allows to conclude.

<figure><img src="image_9.png" /></figure>

Let us now combine all previous controls. We apply the second result of Proposition 12.1, Proposition 12.2 with $Tx/2$ instead of $x$, Proposition 12.4, and Proposition 12.5 with $Tx/2$ instead of $x$. Thus, for each $\varphi$, we have that

$$|G_{\varphi\varphi'} - \mathbb{E}[G_{\varphi\varphi'}]| \leq C^*Te^{-c^*n} + Tx/2 + C^*Te^{-c^*n} + Tx/2 = Tx + C^*Te^{-c^*n}$$

with probability at least

$$1 - C^*\tau e^{-c^*n} - \frac{C_q^*\tau n^q}{T^q x^q} - 4 \exp\left(-\frac{c_q^*T^2x^2}{\tau n^2}\right) - \frac{C_q^*n^q}{T^q x^q}.$$
First since $T \ge 2n\tau$, we have that $\frac{C_q^* \tau n^q}{T q x^q} \le \frac{C_q^*}{\tau^{q-1} x^q}$. Then we also have

$$ \begin{aligned} \exp\left(-\frac{c_q^* T^2 x^2}{\tau n^2}\right) &\le \exp(-c_q^* \tau x^2) \\ &\le \frac{C_q^*}{\tau^{q-1} x^{2(q-1)}} \end{aligned} $$

where we used the fact that $y^s e^{-cy} \le (s/ec)^s$ for $y, c, s > 0$. Finally, we have that $\frac{C_q^* n^q}{T q x^q} \le \frac{C_q^*}{\tau^{q-1} x^q} \le \frac{C_q^*}{\tau^{q-1} x^q}$ which concludes the proof of Lemma 7.12.

## 12.3. Proof of Theorem 7.14 and Proposition 7.15

The proof of Theorem 7.14 is based on Lemma 7.12 and Proposition 7.11 and consists in gathering all previous bounds.

**Proof of Theorem 7.14.** Let us start by defining $n, \tau$ and $r$ to apply Lemma 7.12. Let

$$ n = \left\lfloor \left(\frac{T}{2}\right)^{1/2} \right\rfloor, \quad \tau = \left\lfloor \frac{T}{2n} \right\rfloor, \quad r = T - 2n\tau. $$

Since $T \ge 8(A+1)^2$ we have, $n \ge 2A+1$ and since $T \ge 2$ we have $n \le \sqrt{T/2} \le T/2$ ensuring $\tau \ge 1$. We also have,

$$ n = \left\lfloor (T/2)^{1/2} \right\rfloor \ge \frac{1}{2} \left(\frac{T}{2}\right)^{1/2} $$

and

$$ \tau = \left\lfloor \frac{T}{2n} \right\rfloor \ge \left\lfloor \frac{T}{2(T/2)^{1/2}} \right\rfloor \ge \left\lfloor \left(\frac{T}{2}\right)^{1/2} \right\rfloor \ge \frac{1}{2} \left(\frac{T}{2}\right)^{1/2}. $$

We also have $n \le (T/2)^{1/2}$ and $\tau \le (2T)^{1/2}$. Let us now apply Lemma 7.12 for each pair $(\varphi, \varphi') \in \Phi^2$ with $x = \frac{\alpha \eta_0}{2s(c+1)^2}$. We obtain that,

$$ (12.8) \qquad \forall \varphi, \varphi' \in \Phi, \quad |G_{\varphi\varphi'} - \mathbb{E}[G_{\varphi\varphi'}]| \le Tx + TC^*e^{-c^*n} $$

with probability at least

$$ (12.9) \qquad 1 - \mathrm{Card}(\Phi)^2 C^* \tau e^{-c^*n} - \mathrm{Card}(\Phi)^2 \frac{C_q^*}{\tau^{q-1}} \left( \frac{1}{x^q} + \frac{1}{x^{2(q-1)}} \right). $$

Since $T^{1/2} \ge C^* \log^2(C^*(c+1)^2 s/(\alpha \eta_0))$, we have that $C^* e^{-c^*n} \le x$ and thus (12.8) rewrites as follows,

$$ |G_{\varphi\varphi'} - \mathbb{E}[G_{\varphi\varphi'}]| \le 2Tx = T \frac{\alpha\eta_0}{s(c+1)^2}. $$

Let us look at $\tau e^{-c^*n}$. We have

$$ \begin{aligned} \tau e^{-c^*n} &\le 2^{1/2} T^{1/2} e^{-(c^*/2\sqrt{2})T^{1/2}} \\ &= \frac{1}{T^{(q-1)/2}} 2^{1/2} T^{q/2} e^{-(c^*/2\sqrt{2})T^{1/2}} \\ &\le \frac{C_q^*}{T^{(q-1)/2}} \end{aligned} $$

where we used the same argument $y^s e^{-cy} \le (s/ec)^s$ for $y, c, s > 0$ as at the end of the proof of Lemma 7.12. Since $q \le 2(q-1)$ we have

$$ (12.10) \qquad \frac{1}{x^q} + \frac{1}{x^{2(q-1)}} \le 2 \left(1 + \frac{1}{x^{2(q-1)}}\right), $$
and (12.9) rewrites

$$1 - \mathrm{Card}(\mathbf{\Phi})^2 \frac{C_q^*}{T^{(q-1)/2}} \left(1 + \frac{1}{x^{2(q-1)}}\right).$$

The change of variable $q \leftarrow (q-1)/2$ leads to, for any $q > 1/2$,

$$\mathbb{P}\left(\forall \varphi, \varphi' \in \mathbf{\Phi}, |G_{\varphi\varphi'} - \mathbb{E}[G_{\varphi\varphi'}]| \le T \frac{\alpha\eta_0}{s(c+1)^2}\right) \ge 1 - \mathrm{Card}(\mathbf{\Phi})^2 \frac{C_q^*}{T^q} \left(1 + \left(\frac{s(c+1)^2}{\alpha\eta_0}\right)^{4q}\right).$$

Finally, Proposition 7.11 concludes the proof.

<figure><img src="image_10.png" /></figure>

Let us finish with the proof of Proposition 7.15

**Proof of Proposition 7.15.** Recall that $X$ is the 1-stationary process $\mathbf{HAR}_{\mathfrak{M}^*}(-\infty)$ from Theorem 3.9. Let $\mathbb{Q}$ the distribution under which on $(-\infty, -A)$ the process $X$ has the same distribution as under $\mathbb{P}$, and such that, on $[-A, \infty)$ the process $X$ is independent of the process on $(-\infty, -A)$ and distributed as follows,

* for $m \in \mathcal{M}$, the point processes $S^m \cap [-A, \infty)$ are independent Poisson processes of rate 1,

* For all $(p, k) \in \mathbb{Q}$ with $k \ge -A$, the chains $W_k^p$ are independent centered gaussian random variables with variance $\sigma_p^2$, also independent of the point processes.

The likelihood $\frac{d\mathbb{P}}{d\mathbb{Q}}$ up to time 1 writes as follows,

$$\mathcal{L}_1 = \exp \left( M(1+A) - \sum_{m \in \mathcal{M}} \int_{-A}^{1} \lambda_s^m ds + \sum_{m \in \mathcal{M}} \int_{-A}^{1} \log(\lambda_s^m) dS_s^m + \sum_{p \in \mathcal{P}} \frac{1}{2\sigma_p^2} \int_{-A}^{1} (W_k^p)^2 - (W_k^p - \bar{W}_k^p)^2 d\mathcal{D}_k^p \right)$$

where $\bar{W}_k^p = \Psi^p(\mathfrak{M}^*, t, X_{k-A,k}) = \mathbb{P}^p - \mathbb{P}^p$. Define the following event,

$$\Omega_B = \bigcap_{m \in \mathcal{M}} (S^m([-2A, 1]) \le B) \cap \bigcap_{p \in \mathcal{P}} (\int_{-2A}^{1} |W_k^p| d\mathcal{D}_k^p \le B).$$

Under the distribution $\mathbb{Q}$ it is clear that there exist constants $C^*, c^*$ such that $\mathbb{P}_{\mathbb{Q}}(\Omega_B) \ge 1 - C^*e^{-c^*B}$. Indeed, on the interval $[-2A, -A)$ we have exponential moment by Corollary 5.5 and on $[-A, 1]$ we have independent Poisson processes of rate 1 and independent gaussian variables. Since the interaction functions are bounded and since $\min_{m \in \mathcal{M}} \mu_m > 0$ it is clear that on the event $\Omega_B$ we have

$$\mathcal{L}_1 \ge e^{-C^* B^2} \quad \text{on } \Omega_B,$$

with $C^*$ a constant depending only on $\mathfrak{M}$. Let $x$ such that $|x_{J^c}|_1 \le c|x_J|$ and without loss of generality suppose that $|x|_2 = 1$. Then since $x \cdot G_0^1 x \ge 0$ we have

$$
\begin{align*}
\mathbb{E}[x \cdot G_0^1 x] &= \mathbb{E}_{\mathbb{Q}}[x \cdot G_0^1 x \times \mathcal{L}_1] \\
&\ge \mathbb{E}_{\mathbb{Q}}[x \cdot G_0^1 x \times \mathcal{L}_1 1_{\Omega_B}] \\
&\ge C^* e^{-C^* B^2} \mathbb{E}_{\mathbb{Q}}[x \cdot G_0^1 x 1_{\Omega_B}] \\
&= C^* e^{-C^* B^2} \left(x \cdot \mathbb{E}_{\mathbb{Q}}[G_0^1] x - x \cdot \mathbb{E}_{\mathbb{Q}}[G_0^1 1_{\Omega_B}] x\right) \\
&\ge C^* e^{-C^* B^2} \left(\eta - \|\mathbb{E}_{\mathbb{Q}}[G_0^1 1_{\Omega_B}]\|_{\infty} |x|_1^2\right) \\
&\ge C^* e^{-C^* B^2} \left(\eta - s(c+1)^2 \max_{\varphi, \varphi'} \mathbb{E}_{\mathbb{Q}}[(G_0^1)_{\varphi, \varphi'}^2]^{1/2} \mathbb{P}_{\mathbb{Q}}(\Omega_B^c)^{1/2}\right) \\
&\ge C^* e^{-C^* B^2} \left(\eta - C^* s(c+1)^2 e^{-c^* B}\right)
\end{align*}
$$
where we used Cauchy Schwarz inequality to pass from line 5 to line 6. To conclude the proof one can remark that $E[G] = T E[G_0^1]$. $\square$

## Appendix A: Well posedness and stochastic intensity

In this appendix we prove that $d\pi - dtdx$ is a $(\mathcal{F}_t)_t$ martingale even if $\mathcal{F}_t$ is bigger than the natural history of $\pi$ since it also contains information about the random drifts $\xi$.

Let $(\Omega_1, A_1, P_1)$ a probability space, $\pi$ a Poisson random measure on $\mathbb{R}^2$ on this space and the natural filtration

$$ \mathcal{F}_t^1 = \sigma(\pi \cap (-\infty, t] \times \mathbb{R})). $$

Let $(\Omega_2, A_2, P_2)$ another probability space and $(Z_t)_{t \in \mathbb{R}}$ a process (the random drifts $\xi$ in HAR framework) and denote by $\mathcal{F}^2$ the natural history of $Z$. Consider $(\Omega, \mathcal{F}, \mathbb{P})$ the probability space defined by

$$ \Omega = \Omega_1 \times \Omega_2, \quad \mathcal{F} = A_1 \otimes A_2, \quad \text{and} \quad \mathbb{P} = P_1 \otimes P_2. $$

Then $\pi$ can be viewed as a random variable on $\Omega$ by $\pi(\omega_1, \omega_2) = \pi(\omega_1)$ for any $(\omega_1, \omega_2) \in \Omega$, and similarly for $Z$. In $\Omega$ it is clear that $\pi$ and $Z$ are independent. Consider the filtration

$$ \mathcal{F}_t = \sigma((Z_s)_{s \le t}; \pi \cap (-\infty, t] \times \mathbb{R})). $$

It is clear that we have $\mathcal{F}_t = \mathcal{F}_t^1 \otimes \mathcal{F}_t^2$.

Denote by $\Pi$ the predictable $\sigma$-algebra for $(\mathcal{F}_t)_t$, i.e. the $\sigma$-algebra generated by sets of the form $(s, t] \times B \times A$ with $s < t$, $B$ a borelian set or $\mathbb{R}$ and $A \in \mathcal{F}_s$. Similarly, let $\Pi^1$ the predictable $\sigma$-algebra for $(\mathcal{F}_t^1)_t$.

Let

$$ X : (t, x, \omega_1, \omega_2) \in \mathbb{R} \times \mathbb{R}_+ \times \Omega \mapsto X_t(x, \omega_1, \omega_2) \in \mathbb{R} $$

be a $(\mathcal{F}_t)_t$ predictable process, which means that $X$ is $\Pi$ measurable. We want to show that for any $\omega_2 \in \Omega_2$, the process

$$ X^{\omega_2} : (t, x, \omega_1) \in \mathbb{R} \times \mathbb{R}_+ \times \Omega_1 \mapsto X_t(x, \omega_1, \omega_2) \in \mathbb{R} $$

is $(\mathcal{F}_t^1)_t$ predictable, ie $\Pi^1$ measurable.

Denote by $\mathcal{G}$ the class of processes which satisfy the above property. Clearly for any $s < t$, $B$ borelian, $A_1 \in \mathcal{F}_s^1$ and $A_2 \in \mathcal{F}_s^2$, the following process is in $\mathcal{G}$,

$$ \mathbb{1}_{(s,t] \times B \times A_1 \times A_2}. $$

Since $\mathcal{G}$ is stable by linear combinations and by limits, and because $\mathcal{F}_t = \mathcal{F}_t^1 \otimes \mathcal{F}_t^2$, it is clear that all the processes of the form

$$ \mathbb{1}_{(s,t] \times B \times A}, $$

with $s < t$, $B$ borelian, $A \in \mathcal{F}_s$ are in $\mathcal{G}$. Thus for any $P \in \Pi$, we have $\mathbb{1}_P \in \mathcal{G}$. Then it is clear that $\mathcal{G}$ is exactly the $(\mathcal{F}_t)_t$ predictable processes. We proved that for all $\omega_2 \in \Omega_2$, the process $X^{\omega_2}$ is $(\mathcal{F}_t^1)_t$ predictable.

Thus, if $X$ is non negative, by Fubini Theorem we have

$$ \begin{aligned} E \left[ \int_{\mathbb{R}^2} X(t,x) d\pi(t,x) \right] &= \int_{\Omega} \int_{\mathbb{R}^2} X(t,x,\omega) d\pi(t,x) d\mathbb{P}(\omega) \\ &= \int_{\Omega_2} \left[ \int_{\Omega_1} \int_{\mathbb{R}^2} X^{\omega_2}(t,x,\omega_1) d\pi(t,x) dP_1(\omega_1) \right] dP_2(\omega_2) \\ &= \int_{\Omega_2} \left[ \int_{\Omega_1} \int_{\mathbb{R}^2} X^{\omega_2}(t,x,\omega_1) dt dx dP_1(\omega_1) \right] dP_2(\omega_2) \\ &= \int_{\Omega} \int_{\mathbb{R}^2} X(t,x,\omega) dt dx d\mathbb{P}(\omega) \\ &= E \left[ \int_{\mathbb{R}^2} X(t,x) dt dx \right]. \end{aligned} $$
Thus by taking $X(t, x) = 1_{0 \le x \le \lambda_t}$ with $\lambda$ a $(\mathcal{F}_t)_t$ predictable process, it is clear that the $(\mathcal{F}_t)_t$ compensator of

$$t \mapsto \pi(\{t\} \times [0, \lambda_t]) \text{ is } \lambda_t dt.$$

With this result in hand, we can prove that point processes of HAR processes are well defined if the intensity is defined $dP(\omega) \otimes dt$ almost everywhere.

Let $\lambda$ and $\Upsilon$ two predictable processes, which stand for the intensity in HAR framework. Denote $S_\lambda$ and $S_\Upsilon$ the points under respectively $\lambda$ and $\Upsilon$. It means that

$$S_{\lambda} = \{s \in \mathbb{R} \mid \exists 0 \le x \le \lambda_s, (s, x) \in \pi\}.$$

Suppose that $dP(\omega) \otimes dt$ a.e. we have $\lambda = \Upsilon$. Then we have $S_\lambda = S_\Upsilon$ almost surely. Indeed,

$$\begin{aligned}
\mathbb{E} \left[ \int_{\mathbb{R} \times \mathbb{R}_+} 1_{\min(\lambda_s, \Upsilon_s) < x \le \max(\lambda_s, \Upsilon_s)} d\pi(s, x) \right] &= \mathbb{E} \left[ \int_{\mathbb{R} \times \mathbb{R}_+} 1_{\min(\lambda_s, \Upsilon_s) < x \le \max(\lambda_s, \Upsilon_s)} ds dx \right] \\
&\le \mathbb{E} \left[ \int_{\mathbb{R} \times \mathbb{R}_+} 1_{\lambda_s \ne \Upsilon_s} ds dx \right] \\
&= \int_{\mathbb{R} \times \mathbb{R}_+} \left[ \int_{\Omega \times \mathbb{R}} 1_{\lambda_s \ne \Upsilon_s} dP(\omega) \otimes ds \right] dx \\
&= 0.
\end{aligned}$$

Thus it follows that $S_\lambda = S_\Upsilon$ almost surely.

## Appendix B: Linear HAR processes

In this appendix we present two additional results useful to the study of linear HAR processes. First, in Lemma B.1, we derive $L^1$ bounds and analytic properties of the coefficients defined in Lemma 4.3. Then we extend the use of the coefficients to random sequences in Proposition B.2.

**Lemma B.1.** Let some parameters $\mathfrak{M} \in \mathcal{L}$. Then the coefficients $\left( \left\langle \frac{p,k}{p',k'} \right\rangle_{\mathfrak{M}} \right)_{(p,k), (p',k')} \in \mathcal{Q}, k \ge k'$ from Lemma 4.3 satisfy the following properties.

1. If assumption (Spec1.1) $[\mathfrak{M}]$ holds, then for any $(p_0, k_0) \in \mathcal{Q}$ for any $p \in \mathbf{P}$ we have,

$$\int_{-\infty}^{k_0} \left\langle \begin{array}{cc} p_0, k_0 \\ p, k \end{array} \right\rangle_{\mathfrak{M}} d\mathcal{D}_k^p \le \left[ (\mathrm{Id} - H_W^W)^{-1} \right]_{p_0, p}.$$

2. If assumption (Spec1.1) $[\mathfrak{M}]$ holds, then for any $p_0, p \in \mathbf{P}$ we have,

$$\sum_{k_0 \in \mathcal{D}_{p_0} \cap [0,1)} \int_{k_0}^{\infty} \left\langle \begin{array}{cc} p, k \\ p_0, k_0 \end{array} \right\rangle_{\mathfrak{M}} d\mathcal{D}_k^p \le n_p \times \left[ (\mathrm{Id} - H_W^W)^{-1} \right]_{p, p_0}.$$

3. For non negative sequence $(Z_k^p)_{(p,k) \in \mathcal{Q}}$, the sequence $V$ defined by

$$V_k^p = \sum_{p' \in \mathbf{P}} \int_{-\infty}^{k} \left\langle \begin{array}{cc} p, k \\ p', k' \end{array} \right\rangle_{\mathfrak{M}} Z_{k'}^{p'} d\mathcal{D}_{k'}^{p'}$$

satisfies

$$V_k^p = Z_k^p + \sum_{p' \in \mathbf{P}} \int_{-\infty}^{k-} \alpha_{p'}^p h_{p'}^p (k-k') V_{k'}^{p'} d\mathcal{D}_{k'}^{p'}.$$
The same is true for sequences $(Z_k^p)_{(p,k)\in\mathbb{Q}}$ such that for all $(p_0, k_0) \in \mathbb{Q}$ the following series is absolutely convergent,

$$ \sum_{(p,k) \in \mathbb{Q}, k < k_0} \left[ \sum_{p' \in \mathcal{P}} \int_k^{k_0-} a_{p'}^{p_0} h_{p'}^{p_0} (k_0-k') \langle \begin{smallmatrix} p', k' \\ p, k \end{smallmatrix} \rangle_{\mathfrak{M}} Z_k^p d\mathcal{D}_{k'}^{p'} \right] $$

which is the case if $Z$ is bounded and if assumption (Spec1.1) $[\mathfrak{M}]$ holds.

Point 1 and point 2 give bounds on, respectively, the ascending and descending sum of the coefficients in term of the matrix $H_W^W$. Point 3 extends the use of the coefficients to sequences infinite on both sides.

The following result extends the use of coefficients to random sequences and also deals with uniqueness.

**Proposition B.2.** Let $\mathfrak{M} \in \mathcal{L}$ and suppose that (Spec1.1) $[\mathfrak{M}]$ holds. Suppose that $V$ is a sequence of random variable such that $\sup_{(p,k)\in\mathbb{Q}} \mathbb{E}[|V_k^p|] < \infty$ and satisfying

$$ V_k^p = \sum_{p' \in \mathcal{P}} \int_{-\infty}^{k-} a_{p'}^p h_{p'}^p (k-k') V_{k'}^{p'} d\mathcal{D}_{k'}^{p'}. $$

Then, we must have $V_k^p = 0$ for all $(p, k) \in \mathbb{Q}$. Therefore, for any sequence $Z$ with $\sup_{(p,k)\in\mathbb{Q}} \mathbb{E}[|Z_k^p|] < \infty$, the equation

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

has a unique solution among sequences $V$ for which $\sup_{(p,k)\in\mathbb{Q}} \mathbb{E}[|V_k^p|] < \infty$, being

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

**Proof of Lemma B.1.** We start with **point 1**. Let for $p_0, p \in \mathcal{P}$ and $\tau \ge 1$,

$$ N_{p_0, p}(\tau) := \sup_{k_0 \in \mathcal{D}_{p_0}} \int_{k_0 - \tau}^{k_0} \langle \begin{smallmatrix} p_0, k_0 \\ p, k \end{smallmatrix} \rangle_{\mathfrak{M}} d\mathcal{D}_k^p. $$

Let $k_0 \in \mathcal{D}_{p_0}$, then by point 3 of Lemma 4.3 we have,

$$
\begin{align*}
\int_{k_0 - \tau}^{k_0} \langle \begin{smallmatrix} p_0, k_0 \\ p, k \end{smallmatrix} \rangle_{\mathfrak{M}} d\mathcal{D}_k^p &= 1_{p=p_0} + \int_{k_0 - \tau}^{k_0 -} \langle \begin{smallmatrix} p_0, k_0 \\ p, k \end{smallmatrix} \rangle_{\mathfrak{M}} d\mathcal{D}_k^p \\
&= 1_{p=p_0} + \int_{k_0 - \tau}^{k_0 -} \left[ \sum_{p' \in \mathcal{P}} \int_k^{k_0 -} a_{p'}^{p_0} h_{p'}^{p_0} (k_0 - k') \langle \begin{smallmatrix} p', k' \\ p, k \end{smallmatrix} \rangle_{\mathfrak{M}} d\mathcal{D}_{k'}^{p'} \right] d\mathcal{D}_k^p \\
&= 1_{p=p_0} + \sum_{p' \in \mathcal{P}} \int_{k_0 - \tau}^{k_0 -} \int_{k_0 - \tau}^{k'} a_{p'}^{p_0} h_{p'}^{p_0} (k_0 - k') \langle \begin{smallmatrix} p', k' \\ p, k \end{smallmatrix} \rangle_{\mathfrak{M}} d\mathcal{D}_k^p d\mathcal{D}_{k'}^{p'} \\
&\le 1_{p=p_0} + \sum_{p' \in \mathcal{P}} \int_{k_0 - \tau}^{k_0 -} a_{p'}^{p_0} h_{p'}^{p_0} (k_0 - k') N_{p',p}(\tau) d\mathcal{D}_{k'}^{p'} \\
&\le 1_{p=p_0} + \sum_{p' \in \mathcal{P}} a_{p'}^{p_0} \|h_{p'}^{p_0}\|_{1, n_{p'}}^{\infty} N_{p',p}(\tau).
\end{align*}
$$

Taking the supremum on $k_0$ gives

$$ N_{p_0, p}(\tau) \le 1_{p=p_0} + \sum_{p' \in \mathcal{P}} a_{p'}^{p_0} \|h_{p'}^{p_0}\|_{1, n_{p'}}^{\infty} N_{p', p}(\tau). $$
Thus if $N(\tau) = (N_{p,p'}(\tau))_{p,p' \in \mathcal{P}}$ we have (one should recall that $L_p = 1$)

$$N(\tau) \preceq \mathrm{Id} + H_W^W N(\tau).$$

Equivalently,

$$(\mathrm{Id} - H_W^W) N(\tau) \preceq \mathrm{Id}.$$

Since all the entries of $(\mathrm{Id} - H_W^W)^{-1} = \mathrm{Id} + H_W^W + (H_W^W)^2 + \cdots$ are non negative, one can multiply both sides by $(\mathrm{Id} - H_W^W)^{-1}$ leading to

$$N(\tau) \preceq (\mathrm{Id} - H_W^W)^{-1},$$

and so

$$N_{p_0,p}(\tau) \leq [(\mathrm{Id} - H_W^W)^{-1}]_{p_0,p}.$$

Letting $\tau \to \infty$ gives the intended result. **Point 2** is a consequence of **point 1** and 1-periodicity. Indeed,

$$
\begin{aligned}
& \sum_{k_0 \in \mathcal{D}_{p_0} \cap [0,1)} \int_{k_0}^{\infty} \left\langle \begin{array}{cc} p, k \\ p_0, k_0 \end{array} \right\rangle_{\mathfrak{M}} d\mathcal{D}_k^p \\
&= \sum_{k_0 \in \mathcal{D}_{p_0} \cap [0,1)} \int_{k_0}^{\infty} \left\langle \begin{array}{cc} p, k - \lfloor k \rfloor \\ p_0, k_0 - \lfloor k \rfloor \end{array} \right\rangle_{\mathfrak{M}} d\mathcal{D}_k^p \\
&= \sum_{k_1 \in \mathcal{D}_p \cap [0,1)} \sum_{k_0 \in \mathcal{D}_{p_0} \cap [0,1)} \int_{k_0}^{\infty} \left\langle \begin{array}{cc} p, k_1 \\ p_0, k_0 - \lfloor k \rfloor \end{array} \right\rangle_{\mathfrak{M}} \mathbb{1}_{k - \lfloor k \rfloor = k_1} d\mathcal{D}_k^p \\
&= \sum_{k_1 \in \mathcal{D}_p \cap [0,1)} \sum_{k_0 \in \mathcal{D}_{p_0} \cap [0,1]} \sum_{\mathfrak{Z} \ni j \geq k_0 - k_1} \left\langle \begin{array}{cc} p, k_1 \\ p_0, k_0 - j \end{array} \right\rangle_{\mathfrak{M}} \\
&= \sum_{k_1 \in \mathcal{D}_p \cap [0,1]} \int_{-\infty}^{k_1} \left\langle \begin{array}{cc} p, k_1 \\ p_0, k \end{array} \right\rangle_{\mathfrak{M}} d\mathcal{D}_k^{p_0}.
\end{aligned}
$$

Here **point 1** concludes the proof of **point 2** since there $\mathrm{Card}(\mathcal{D}_p \cap [0,1)) = n_p$. Let us tackle **point 3**. From point 3 of Lemma 4.3 we have (the assumptions allow us to use Fubini Theorem),

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

Which is the result. Finally, if $Z$ is bounded, say by $Z_{\infty} < \infty$ and $\mathrm{SpR}(H_W^W) < 1$ then,

$$\sum_{(p,k) \in \mathcal{Q}, k < k_0} \left[ \sum_{p' \in \mathcal{P}} \int_k^{k_0-} a_{p'}^{p_0} h_{p'}^{p_0} (k_0 - k') \left\langle \begin{array}{cc} p', k' \\ p, k \end{array} \right\rangle_{\mathfrak{M}} Z_k^p d\mathcal{D}_{k'}^{p'} \right]$$
$$
\begin{align*}
&\le Z_\infty \sum_{p \in \mathcal{P}} \int_{-\infty}^{k_0-} \sum_{p' \in \mathcal{P}} \int_k^{k_0-} a_{p'}^{p_0} h_{p'}^{p_0} (k_0 - k') \begin{pmatrix} p', k' \\ p, k \end{pmatrix}_{\mathfrak{M}} d\mathcal{D}_{k'}^{p'} d\mathcal{D}_k^{p'} \\
&= Z_\infty \sum_{p' \in \mathcal{P}} \int_{-\infty}^{k_0-} \sum_{p \in \mathcal{P}} \int_{-\infty}^{k'} a_{p'}^{p_0} h_{p'}^{p_0} (k_0 - k') \begin{pmatrix} p', k' \\ p, k \end{pmatrix}_{\mathfrak{M}} d\mathcal{D}_k^{p'} d\mathcal{D}_{k'}^{p'} \\
&\le Z_\infty \max_{p'} \left( \sum_{p \in \mathcal{P}} [(\mathrm{Id} - H_W^W)^{-1}]_{p',p} \right) \sum_{p' \in \mathcal{P}} \int_{-\infty}^{k_0-} a_{p'}^{p_0} h_{p'}^{p_0} (k_0 - k') d\mathcal{D}_{k'}^{p'} \\
&\le Z_\infty \max_{p'} \left( \sum_{p \in \mathcal{P}} [(\mathrm{Id} - H_W^W)^{-1}]_{p',p} \right) \sum_{p' \in \mathcal{P}} a_{p'}^{p_0} \|h_{p'}^{p_0}\|_{1,n_{p'}}^\infty < \infty.
\end{align*}
$$

Where we use **point 1** between the third and the fourth line.

<figure><img src="image_11.png" /></figure>

**Proof of Proposition B.2.** Let $\mathcal{V}_\infty^p = \sup_{k \in \mathcal{D}_p} \mathbb{E}[|V_k^p|]$ for $p \in \mathcal{P}$. Then for any $p \in \mathcal{P}$ we have,

$$
\mathbb{E}[|V_k^p|] \le \sum_{p' \in \mathcal{P}} \int_{-\infty}^{k-} a_{p'}^p h_{p'}^p (k-k') \mathbb{E}[|V_{k'}^{p'}|] d\mathcal{D}_{k'}^{p'} \le \sum_{p' \in \mathcal{P}} (H_W^W)_{pp'} \mathcal{V}_\infty^{p'}.
$$

Thus, by taking a supremum on $k \in \mathcal{D}_p$,

$$
\mathcal{V}_{\infty}^{p} \leq \sum_{p' \in \mathcal{P}} (H_{W}^{W})_{pp'} \mathcal{V}_{\infty}^{p'} = (H_{W}^{W} \mathcal{V}_{\infty})_{p}.
$$

Iterating this formula gives $\mathcal{V}_\infty \le (H_W^W)^n \mathcal{V}_\infty \xrightarrow{n \to \infty} \mathbf{0}_P$ which implies $\mathcal{V}_\infty = \mathbf{0}_P$. Which proves the first result.

Finally, since $Z$ is bounded in $L^1$ the series in point 3 of Lemma B.1 is absolutely convergent in expectation and thus it is almost surely absolutely convergent. It means that the random sequence $Y$ is a solution of the equation and it is bounded in $L^1$ since

$$
\sup_{(p,k) \in \mathcal{Q}} \mathbb{E} \left[ \left| \sum_{p' \in \mathcal{P}} \int_{-\infty}^{k-} \begin{pmatrix} p, k \\ p', k' \end{pmatrix}_{\mathfrak{M}} Z_{k'}^{p'} d\mathcal{D}_{k'}^{p'} \right| \right] \le \mathcal{V}_{\infty} \max_{p \in \mathcal{P}} \sum_{p' \in \mathcal{P}} [(\mathrm{Id} - H_W^W)^{-1}]_{p,p'} < \infty
$$

where $\mathcal{F}_\infty = \sup_{(p,k) \in \mathcal{Q}} \mathbb{E}[|Z_k^p|]$.

<figure><img src="image_12.png" /></figure>

**Appendix C: Exponential moments**

In this appendix we state and prove the key result to derive exponential moments for HAR processes. Lemma C.1 states exponential moments for linear combinations of both the random drifts and point processes with clusters generated by the cluster functions defined in Definition 4.7 and immigrant rates depending on linear functionals of the random drifts. Recall that the random drifts $(\xi_k^p)_{(p,k) \in \mathcal{Q}}$ are independent random real variables such that for all $(p,k) \in \mathcal{Q}$ we have $\xi_k^p \sim \xi_{k+n}^p$ for any $n \in \mathbb{Z}$.

Consider the random variable

$$
\mathcal{Z} = \sum_{m \in \mathcal{M}} \int_{\mathbb{R}} B^m(t) dS_t^m + \sum_{p \in \mathcal{P}} \int_{\mathbb{R}} C_k^p \xi_k^p d\mathcal{D}_k^p =: B \bullet S + C \bullet \xi,
$$

where we have

• $(S^m)_{m \in \mathcal{M}}$ a multivariate point process such that conditionally to the $(\xi_k^p)_{(p,k) \in \mathcal{Q}}$ it has immigrant rates being

$$
\mathcal{I}_t^m = i_m + \sum_{(p,k) \in \mathcal{Q}} A_{p,k}^m(t) \xi_k^p
$$
and independent clusters generated by the bivariate functions $(\mathfrak{h}_{m'}^m)_{m,m' \in M}$. In other words, the intensity $\lambda_t^m$ of $S^m$ at time $t$ is

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

* $i := (i_m)_{m \in M}$ is a non negative vector.

* $A_{p,k}^m$ is a non negative real function for all $m \in M$, $(p, k) \in Q$. We also define the following matrix

$$ A_{\infty} = \left( \sup_{t \in \mathbb{R}} \int_{\mathbb{R}} A_{p,k}^m(t) d\mathcal{D}_k^p \right)_{(p,m) \in \mathcal{P} \times \mathcal{M}}. $$

* $B = (B^m)_{m \in M}$ is a vector of non negative real functions and for a positive integer $n$ we define the vector

$$ B_{\infty, n} = \left( \|B^m\|_{1,1/n}^{\infty} \right)_{m \in M}. $$

* $C = (C^p)_{p \in \mathcal{P}}$ is a vector of non negative functions on the $\mathcal{D}_p$'s and we define the vector

$$ C_{\infty} = \left( \int_{\mathbb{R}} C_k^p d\mathcal{D}_k^p \right)_{p \in \mathcal{P}}. $$

We can now state the main result.

**Lemma C.1.** Let $n \ge 1$ an integer. Suppose that assumption (SubG) $[\xi]$ holds. Then we have

$$ \mathbb{E} [e^{\mathcal{Z}}] \le \exp \left[ t^{\mathcal{T}} (ne^{\mathcal{L}(B_{\infty,n})} - n\mathbf{1}) + \mathcal{E}^{\mathcal{T}} \mathcal{X} + \mathcal{X}^{\mathcal{T}} \mathrm{diag}(\mathbf{s}) \mathcal{X} \right] $$

where

$$ \mathcal{X} = C_{\infty} + A_{\infty}(ne^{\mathcal{L}(B_{\infty,n})} - n\mathbf{1}), $$

with $\mathcal{L}$ the log-Laplace function of a $\mathcal{P}$ois($\mathfrak{H}_{\mathfrak{m}}^{\mathcal{T}}$) Galton Watson tree defined in Theorem 3.1 of [41].

**Proof of Lemma C.1.** By conditioning on the random drifts we obtain

$$ \begin{aligned} \mathbb{E} [e^{\mathcal{Z}}] &= \mathbb{E} [\mathbb{E} [e^{\mathcal{Z}} | \xi]] \\ &= \mathbb{E} [e^{C \cdot \xi} \mathbb{E} [e^{B \cdot S} | \xi]] \end{aligned} $$

Let us investigate the term $\mathbb{E} [e^{B \cdot S} | \xi]$. We will use the cluster decomposition. Let $G_t^m$ a $\mathcal{P}$ois($\mathfrak{h}^{\mathcal{T}}$) born at time $t \in \mathbb{R}$ with root of type $m \in M$ and let

$$ G_t^m(B) := \sum_{(u,m',s) \in G_t^m} B^{m'}(s) $$

the contribution of cluster $G_t^m$ to $B \cdot S$. From the cluster decomposition of $S$ we have

$$ \mathbb{E} [e^{B \cdot S} | \xi] = \exp \left[ \sum_{m \in \mathcal{M}} \int_{\mathbb{R}} (\psi^m(t, B) - 1) \mathcal{I}_t^m dt \right] $$

where

$$ \psi^m(t, B) = \mathbb{E} [e^{G_t^m(B)}]. $$

We can now use the expression of the immigrant rates $\mathcal{I}^m$,

$$ \mathbb{E} [e^{B \cdot S} | \xi] = \exp \left[ \sum_{m \in \mathcal{M}} i_m \int_{\mathbb{R}} (\psi^m(t, B) - 1) dt + \sum_{p \in \mathcal{P}} \int_{\mathbb{R}} \xi_k^p \sum_{m \in \mathcal{M}} \int_{\mathbb{R}} A_{p,k}^m(t) (\psi^m(t, B) - 1) dt d\mathcal{D}_k^p \right] $$
Thus we have

$$ \mathbb{E}[e^{\mathcal{Z}}] = \exp\left[\sum_{m \in \mathcal{M}} i_m \int_{\mathbb{R}} (\psi^m(t, B) - 1)dt\right] \mathbb{E}\left[\exp\left(\sum_{p \in \mathcal{P}} \int_{\mathbb{R}} D_k^p \xi_k^p d\mathcal{D}_k^p\right)\right] $$

with

$$ D_k^p = C_k^p + \sum_{m \in \mathcal{M}} \int_{\mathbb{R}} A_{p,k}^m(t)(\psi^m(t, B) - 1)dt. $$

Thus from assumption (SubG) $[\xi]$ we have

$$ \mathbb{E}[e^{\mathcal{Z}}] \le \exp \left[ \sum_{m \in \mathcal{M}} i_m \int_{\mathbb{R}} (\psi^m(t, B) - 1)dt + \sum_{p \in \mathcal{P}} \mathfrak{e}_p \int_{\mathbb{R}} D_k^p d\mathcal{D}_k^p + \sum_{p \in \mathcal{P}} \mathfrak{s}_p \int_{\mathbb{R}} (D_k^p)^2 d\mathcal{D}_k^p \right] $$

Let us focus on $\int_{\mathbb{R}} D_k^p d\mathcal{D}_k^p$. We have,

$$ \begin{aligned} \int_{\mathbb{R}} D_k^p d\mathcal{D}_k^p &= \int_{\mathbb{R}} \left(C_k^p + \sum_{m \in \mathcal{M}} \int_{\mathbb{R}} A_{p,k}^m(t)(\psi^m(t, B) - 1)dt\right) d\mathcal{D}_k^p \\ &= (C_\infty)_p + \sum_{m \in \mathcal{M}} \int_{\mathbb{R}} \int_{\mathbb{R}} A_{p,k}^m(t) d\mathcal{D}_k^p (\psi^m(t, B) - 1)dt \\ &\le (C_\infty)_p + \sum_{m \in \mathcal{M}} (A_\infty)_{p,m} \int_{\mathbb{R}} (\psi^m(t, B) - 1)dt. \end{aligned} $$

To continue the proof we have to bound $\int_{\mathbb{R}} (\psi^m(t, B) - 1)dt$ for $m \in \mathcal{M}$. Recall that $n$ is a positive integer. We use the same ideas as in Theorem 3.10 of [41] except we have to deal with 1-stationary.

For $t \in \mathbb{R}$, denote by $\lfloor t \rfloor_n$ and $\{t\}_n$ the unique real numbers such that $t = \lfloor t \rfloor_n + \{t\}_n$ with $\lfloor t \rfloor_n \in n\mathbb{Z}$, $0 \le \{t\}_n < n$. Thus we have by 1-periodicity of clusters, if $\tau_x(s) = x + s$, we have

$$ \psi^m(t, B) = \mathbb{E}[e^{G_t^m(B)}] = \psi^m(\{t\}_n, B \circ \tau_{\lfloor t \rfloor_n}). $$

It follows that

$$ \begin{aligned} \int_{\mathbb{R}} (\psi^m(t, B) - 1)dt &= \int_0^n \sum_{j \in \mathbb{Z}} (\psi^m(t + nj, B) - 1)dt \\ &= \int_0^n \sum_{j \in \mathbb{Z}} (\psi^m(t, B \circ \tau_{nj}) - 1)dt \\ &= \int_0^n \mathbb{E} \left[ \sum_{j \in \mathbb{Z}} (e^{G_t^m(B \circ \tau_{nj})} - 1) \right] dt \\ &\le \int_0^n \mathbb{E} \left[ -1 + \prod_{j \in \mathbb{Z}} e^{G_t^m(B \circ \tau_{nj})} \right] dt \\ &= \int_0^n \mathbb{E} \left[ -1 + e^{G_t^m(\sum_{j \in \mathbb{Z}} B \circ \tau_{nj})} \right] dt \\ &\le \int_0^n \mathbb{E} \left[ -1 + e^{G_t^m(B_{\infty,n})} \right] dt \\ &\le \int_0^n \left[ e^{\mathcal{S}(B_{\infty,n})} - 1 \right]_m dt \end{aligned} $$
$$= n[e^{\mathcal{L}(B_{\infty,n})} - \mathbf{1}]_m.$$

The last inequality comes from the fact that $\{(u, m') | (u, m', s) \in G_t^m\}$ is stochastically dominated by a $\mathcal{P}$ois($\mathfrak{H}_{\mathfrak{M}}^\top$) Galton Watson process, and thus, with the notations of Theorem 3.1 of [41], $G_t^m(B_{\infty,n}) \le B_{\infty,n} \cdot \text{Card}_M(\mathcal{T}^m)$ for a well coupled $\mathcal{P}$ois($\mathfrak{H}_{\mathfrak{M}}^\top$) Galton Watson process $\mathcal{T}^m$ with root of type $m$. We conclude that

$$\int_{\mathbb{R}} D_k^p d\mathcal{D}_k^p \le (C_\infty)_p + [A_\infty(ne^{\mathcal{L}(B_{\infty,n})} - n\mathbf{1})]_p.$$

Finally, since $\int_{\mathbb{R}} (D_k^p)^2 d\mathcal{D}_k^p \le (\int_{\mathbb{R}} D_k^p d\mathcal{D}_k^p)^2$ we can conclude that

$$\begin{aligned}
\mathbb{E}[e^{\mathcal{Z}}] &= \exp \left[ i_m \int_{\mathbb{R}} (\psi^m(t, B) - 1)dt + \sum_{p \in \mathcal{P}} \mathfrak{e}_p \int_{\mathbb{R}} D_k^p d\mathcal{D}_k^p + \sum_{p \in \mathcal{P}} \mathfrak{s}_p \int_{\mathbb{R}} (D_k^p)^2 d\mathcal{D}_k^p \right] \\
&\le \exp \left[ \sum_{m \in \mathcal{M}} i_m \int_{\mathbb{R}} (\psi^m(t, B) - 1)dt + \sum_{p \in \mathcal{P}} \mathfrak{e}_p \int_{\mathbb{R}} D_k^p d\mathcal{D}_k^p + \sum_{p \in \mathcal{P}} \mathfrak{s}_p \left( \int_{\mathbb{R}} D_k^p d\mathcal{D}_k^p \right)^2 \right] \\
&\le \exp \left[ i^\top (ne^{\mathcal{L}(B_{\infty,n})} - n\mathbf{1}) + \mathfrak{e}^\top \mathcal{X} + \mathcal{X}^\top \text{diag}(\mathfrak{s}) \mathcal{X} \right]
\end{aligned}$$

where we denoted $\mathcal{X} := C_\infty + A_\infty(ne^{\mathcal{L}(B_{\infty,n})} - n\mathbf{1})$. $\square$

## Appendix D: Asymptotic limit

This appendix is dedicated to Lemma D.1, a technical Lemma used in the proof of Theorem 6.3. Recall that in the proof of Theorem 6.3 we have two HAR processes with different initial conditions, indexed by $i = 1, 2$. The process $((\Delta\lambda^m)_{m \in \mathcal{M}}, (\Delta S^m)_{m \in \mathcal{M}}, (\Delta W^p)_{p \in \mathcal{P}})$ correspond to the difference, in absolute value between these two processes.

We recall that, to avoid confusion, in this proof, we denote by $g$ the interaction functions of $\mathfrak{M}^+$ and $g$ the cluster functions for $\mathfrak{M}^+$.

Lemma D.1 bounds the difference process by a linear HAR process with vanishing base rates.

**Lemma D.1.** Suppose that assumptions (Spec1.1)[$\mathfrak{M}$] and (ExpTail)[$\mathfrak{M}$] hold. Then there exists an adapted process $((\mathcal{L}^m)_{m \in \mathcal{M}}, (\mathcal{S}^m)_{m \in \mathcal{M}}, (\mathcal{W}^p)_{p \in \mathcal{P}})$ and an adapted Poisson random measures $\mathcal{P}^m$ for $m \in \mathcal{M}$ such that it dominates $((\Delta\lambda^m)_{m \in \mathcal{M}}, (\Delta S^m)_{m \in \mathcal{M}}, (\Delta W^p)_{p \in \mathcal{P}})$, which means

* 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 all $m \in \mathcal{M}$, $t \ge t_0$, $C \subset [t_0, \infty)$, $(p, k) \in \mathcal{Q}$ with $k \ge t_0$,

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

where $u_0 = K|\mathcal{C}_1, \mathcal{C}_2|_\delta$ with $K, c, \delta > 0$ constants depending only on $\mathfrak{M}$.
**Proof of Lemma D.1.** Recall that for $i = 1, 2$ we have for $t \ge t_0$, $C \subset \mathbb{R}$ and $k \ge t_0$

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

Let $\mathcal{P}^m$ the Poisson random measure on $[t_0, \infty) \times \mathbb{R}_+$ defined by

$$ \mathcal{P}^m(s, x) = \pi^m(s, x + \min[\lambda_s^m, \mathcal{E}_1, \lambda_s^m, \mathcal{E}_2]). $$

Then it follows that for $C \subset [t_0, \infty)$,

$$ (D.1) \qquad \Delta S^m(C) = \int_{C \times \mathbb{R}_+} \mathbb{1}_{x \le \Delta \lambda^m(s)} d\mathcal{P}^m(s, x). $$

Since $\Phi^m$ is $L_m$ Lipschitz and from triangular inequality,

$$
\begin{aligned}
\Delta \lambda^m(t) &\le \sum_{m' \in \mathcal{M}} \int_{t_0}^{t-} g_{m'}^m (t-s) d\Delta S^{m'}(s) + \sum_{p \in \mathcal{P}} \int_{t_0}^{t-} g_p^m (t-k) a_p^m \Delta W^p(k) d\mathcal{D}_k^p \\
&\quad + \sum_{m' \in \mathcal{M}} \int_{-\infty}^{t_0-} g_{m'}^m (t-s) d(\mathcal{E}_1^{m'} \Delta \mathcal{E}_2^{m'})_s \\
&\quad + \sum_{p \in \mathcal{P}} \int_{-\infty}^{t_0-} g_p^m (t-k) |\kappa_p^m(\mathcal{E}_{1,k}^p) - \kappa_p^m(\mathcal{E}_{2,k}^p)| d\mathcal{D}_k^p
\end{aligned}
$$

with $g_{\beta}^{\alpha} = L_{\alpha}|h_{\beta}^{\alpha}|$. From (ExpTail) [M] there exists a constant $c_1 > 0$ such that $g_{\beta}^{\alpha}(x) \le K_1 e^{-c_1 x}$ thus we have

$$
\begin{aligned}
\sum_{m' \in \mathcal{M}} \int_{-\infty}^{t_0-} g_{m'}^m (t-s) d(\mathcal{E}_1^{m'} \Delta \mathcal{E}_2^{m'})_s &\le K_1 e^{-c_1(t-t_0)} \sum_{m' \in \mathcal{M}} \int_{-\infty}^{t_0-} e^{-c_1(t_0-s)} d(\mathcal{E}_1^{m'} \Delta \mathcal{E}_2^{m'})_s \\
&\le K e^{-c(t-t_0)} |\mathcal{E}_1, \mathcal{E}_2|_{\delta}
\end{aligned}
$$

with $c = \delta = c_1 > 0$ and $K$ depending on $\mathfrak{M}$. The function $\psi$, defined in (6.1) is such that for any Lipschitz function $f$ with the convention $f(\emptyset) = 0$, for any $x, y \in \mathbb{R} \cup \{\emptyset\}$ we have

$$ |f(x) - f(y)| \le (\text{Lip}(f) + |f(0)|)\psi(x, y). $$

Thus we have

$$ |\kappa_p^m(\mathcal{E}_{1,k}^p) - \kappa_p^m(\mathcal{E}_{2,k}^p)| \le K_2 \psi(\mathcal{E}_{1,k}^p, \mathcal{E}_{2,k}^p) $$

where $K_2$ is a constant depending on the masks $\kappa_p^m$. It follows that

$$ \sum_{p \in \mathcal{P}} \int_{-\infty}^{t_0-} g_p^m (t-k) |\kappa_p^m(\mathcal{E}_{1,k}^p) - \kappa_p^m(\mathcal{E}_{2,k}^p)| d\mathcal{D}_k^p \le K e^{-c(t-t_0)} |\mathcal{E}_1, \mathcal{E}_2|_{\delta}. $$

Thus, if $u_0 = K |\mathcal{E}_1, \mathcal{E}_2|_{\delta}$, up to increasing $K$ we have,

$$ (D.2) \qquad \Delta \lambda^m(t) \le u_0 e^{-c(t-t_0)} + \sum_{m' \in \mathcal{M}} \int_{t_0}^{t-} g_{m'}^m (t-s) d\Delta S^{m'}(s) + \sum_{p \in \mathcal{P}} \int_{t_0}^{t-} g_p^m (t-k) a_p^m \Delta W^p(k) d\mathcal{D}_k^p $$
Similarly,

$$ (D.3) \qquad \Delta W^p(k) \le u_0 e^{-c(k-t_0)} + \sum_{m \in \mathcal{M}} \int_{t_0}^{k-} g_m^p(k-s) d\Delta S^m(s) + \sum_{p' \in \mathcal{P}} \int_{t_0}^{k-} g_{p'}^p(k-k') a_{p'}^p \Delta W^{p'}(k') d\mathcal{D}_{k'}^{p'}. $$

Thus a process $((\tilde{\mathcal{L}}^m)_{m \in \mathcal{M}}, (\tilde{\mathcal{S}}^m)_{m \in \mathcal{M}}, (\tilde{\mathcal{W}}^p)_{p \in \mathcal{P}})$ satisfying equality in equations (D.1), (D.2) and (D.3), ie

$$ (D.4) \qquad \left\{ \begin{aligned} \tilde{\mathcal{L}}_t^m &= u_0 e^{-c(t-t_0)} + \sum_{m' \in \mathcal{M}} \int_{t_0}^{t-} g_{m'}^m(t-s) d\tilde{\mathcal{S}}_{s}^{m'} + \sum_{p \in \mathcal{P}} \int_{t_0}^{t-} g_p^m(k-p) a_p^m \tilde{\mathcal{W}}_k^p d\mathcal{D}_k^p \\ \tilde{\mathcal{S}}^m(C) &= \int_{C \times \mathbb{R}_+} \mathbb{1}_{x \le \tilde{\mathcal{L}}_s^m} d\mathcal{P}^m(s, x) \\ \tilde{\mathcal{W}}_k^p &= u_0 e^{-c(k-t_0)} + \sum_{m \in \mathcal{M}} \int_{t_0}^{k-} g_m^p(k-s) d\tilde{\mathcal{S}}_{s}^{m} + \sum_{p' \in \mathcal{P}} \int_{t_0}^{k-} g_{p'}^p(k-k') a_{p'}^p \tilde{\mathcal{W}}_{k'}^{p'} d\mathcal{D}_{k'}^{p'} \end{aligned} \right. $$

must dominate $((\Delta \lambda^m)_{m \in \mathcal{M}}, (\Delta S^m)_{m \in \mathcal{M}}, (\Delta W^p)_{p \in \mathcal{P}})$. Equations (D.4) is in fact linear HAR equations with parameter from $\mathfrak{M}_+$ with $b_p^\alpha = 0$ and vanishing base rates instead of constant base rates for the point processes and random drifts for the chains. Thus one can use the results on linear models. More precisely, we can express exactly the process with the cluster functions $g$ and the coefficients $\langle p, k \rangle_{\mathfrak{M}_+}$ as follows,

$$ \left\{ \begin{aligned} \tilde{\mathcal{L}}_t^m &= u_0 e^{-c(t-t_0)} + \sum_{p, p' \in \mathcal{P}} \int_{t_0}^{t-} \int_{t_0}^{k-} u_0 e^{-c(k'-t_0)} \langle p, k \rangle_{\mathfrak{M}_+} g_p^m(t-k) a_p^m d\mathcal{D}_{k'}^{p'} d\mathcal{D}_k^p + \sum_{m' \in \mathcal{M}} \int_{t_0}^{t-} g_{m'}^m(t-s) d\tilde{\mathcal{S}}_s^{m'} \\ \tilde{\mathcal{S}}^m(C) &= \int_{C \times \mathbb{R}_+} \mathbb{1}_{x \le \tilde{\mathcal{L}}_s^m} d\mathcal{P}^m(s, x) \\ \tilde{\mathcal{W}}_k^p &= \sum_{p' \in \mathcal{P}} \int_{t_0}^{k-} u_0 e^{-c(k'-t_0)} \langle p, k \rangle_{\mathfrak{M}_+} d\mathcal{D}_{k'}^{p'} + \sum_{p' \in \mathcal{P}} \int_{t_0}^{k-} \sum_{m \in \mathcal{M}} \int_{t_0}^{k'-} g_m^p(k'-s) d\tilde{\mathcal{S}}_s^m \langle p, k \rangle_{\mathfrak{M}_+} d\mathcal{D}_{k'}^{p'} \end{aligned} \right. . $$

Then one can use the exponential decay of the coefficients stated in Proposition 6.1, and (ExpTail) $[\mathfrak{M}^+]$ (trivially implied by (ExpTail) $[\mathfrak{M}]$) to prove that we have (up to reducing $c$ and increasing $K$),

$$ (D.5) \qquad \left\{ \begin{aligned} \tilde{\mathcal{L}}_t^m &\le u_0 K e^{-c(t-t_0)} + \sum_{m' \in \mathcal{M}} \int_{t_0}^{t-} g_{m'}^m(t-s) d\tilde{\mathcal{S}}_s^{m'} \\ \tilde{\mathcal{S}}^m(C) &= \int_{C \times \mathbb{R}_+} \mathbb{1}_{x \le \tilde{\mathcal{L}}_s^m} d\mathcal{P}^m(s, x) \\ \tilde{\mathcal{W}}_k^p &\le u_0 K e^{-c(k-t_0)} + \sum_{m \in \mathcal{M}} \int_{t_0}^{k-} K_4 e^{-c(k-s)} d\tilde{\mathcal{S}}_s^m \end{aligned} \right. . $$

The searched process is the one which satisfies equality in (D.5) and then doing $u_0 \leftarrow Ku_0$ concludes. $\square$

## Appendix E: LASSO estimation for HAR processes

The last appendix we present three technical results used in Section 7. The first one bounds the 1-stationary HAR process over all possible sliding windows of size $A$ in the interval $[-A, T)$. The second and third results are respectively, concentration inequalities (generalized Bernstein inequalities) for point processes and gaussian processes.

In our statistical study, considering the support assumption of the interaction functions, we need to control locally the $\text{HAR}_{\mathfrak{M}^*}(-\infty)$ process, denoted $X = (\lambda, S, W)$, on time window of size $A$, over all the time window $[-A, T)$. Informally we have to control $\max_{t \in [0, T)} \|X_{t-A, t}\|_{\infty}$. $\square$
Let for $m \in M$,

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

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

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

Finally, let $Y_T = \max(\max_m Y^m(T), \max_{\alpha,p} Y_p^\alpha(T))$.

**Proposition E.1.** There exists constants $C^*$ and $c^*$ depending on $\mathfrak{M}^*$ (and thus possibly on $A$) such that for all $T \ge 1$ and all $x \ge 0$,

$$\mathbb{P}(Y_T \le x) \ge 1 - C^*Te^{-c^*x}.$$

Thus for all $q > 0$ we have,

$$\mathbb{P}(Y_T \le C^*(q+1)\log(T)) \ge 1 - \frac{C^*}{T^q}.$$

If $\mathbb{Y}_T$ is defined as $Y_T$ but with $\mathbb{X}$, the coupling from Definition 6.5, instead of $X$, then the same holds for $\mathbb{Y}_T$.

To calibrate the LASSO we need controls on the fluctuations of the contrast around its compensator; $b_\varphi - \tilde{b}_\varphi$ for all $\varphi \in \Phi$. These fluctuations are broken into two terms: $J_{\varphi,M}$ and $J_{\varphi,P}$. Recall that for $\varphi \in \Phi$ we have

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

with $\Psi^m(\varphi, t, X_{t-A,t})$ a predictable process.

Lemma E.2 is the analogue, for point processes, of Bernstein inequality. It is an adaptation of [28] and controls the term $J_{\varphi,M}$.

**Lemma E.2.** Let $x > 0$ and $q > 0$. If $\varphi \in \Phi_P \cup \Phi^P$ then we have

$$\mathbb{P}(|J_{\varphi,M}| = 0) = 1.$$

Else, define $B_\varphi$ by

$$B_\varphi = \begin{cases} 1 & \text{if } \varphi \in \Phi_M \\ C^*(q+1) \log(T) & \text{if } \varphi \in \Phi^M \end{cases}$$

and let

$$V_{\varphi,M} = \frac{1}{3-e} \left( B_{\varphi}^2 x + \sum_{m \in M} \int_0^T \Psi^m(\varphi, t, X_{t-A,t})^2 dS_t^m \right).$$

Then we have

$$(E.1) \qquad |J_{\varphi,M}| \le \sqrt{3xV_{\varphi,M}} + \frac{B_{\varphi}x}{3} \le B_{\varphi} \left( \sqrt{\frac{3x(x + C^*(q+1)T \log(T))}{3-e}} + \frac{x}{3} \right)$$

with probability at least

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

The second term, $J_{\varphi,P}$ writes as follow,

$$J_{\varphi,P} = \sum_{p \in P} \int_{0}^{T-} \Psi^p(\varphi, k, X_{k-A,k}) \varepsilon_k^p d\mathcal{D}_k^p.$$
with $\Psi^p(\varphi, k, X_{k-A,k})$ a predictable process and $\varepsilon_k^p$ independent and centered gaussian random variables with variance $\sigma_p^2$.

Lemma E.3 is a Bernstein like inequality but for generalized gaussian processes. It controls the term $J_{\varphi,P}$.

**Lemma E.3.** Let $x > 0$ and $q > 0$. If $\varphi \in \mathbf{\Phi}_M \cup \mathbf{\Phi}^M$ then we have

$$ \mathbb{P}(|J_{\varphi,P}| = 0) = 1. $$

Else, define $B_\varphi$ by

$$ A_{\varphi} = \begin{cases} 1 & \text{if } \varphi \in \mathbf{\Phi}_P \\ C^*(q+1) \log(T) & \text{if } \varphi \in \mathbf{\Phi}^P \end{cases} $$

and let

$$ V_{\varphi,P} = \sum_{p \in \mathbf{P}} \int_{0}^{T} \Psi^{p}(\varphi, k, X_{k-A,k})^{2} \sigma_{p}^{2} d\mathcal{D}_{k}^{p}. $$

Then there exists $C^*$ depending only on $\mathfrak{M}^*$ such that, if $\bar{\sigma} = \max_{p \in \mathbf{P}} \sigma_p$, for all $x > 0$ we have

$$ (E.2) \qquad |J_{\varphi,P}| \le \sqrt{3x(V_{\varphi,P} + B_{\varphi}^2 \bar{\sigma}^2)} \le B_{\varphi} \sqrt{C^* x T} $$

with probability at least

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

**Proof of Proposition E.1.** Remark that $Y_T$ is an increasing function of $A$ so we can suppose that $A \ge 1$. To have only a maximum over a finite number of random variables we can notice that

$$ Y_T \le 2 \max_{n=-1, \dots, \lfloor T/A \rfloor} \max_{m, \alpha, p} \left( S^m([nA, nA + A]), \int_{nA}^{(nA+A)-} |\kappa_p^\alpha(W_k^p)| d\mathcal{D}_k^p \right) $$

From the exponential moments we have,

$$ \mathbb{P}(S^m([nA, nA + A]) \ge x) \le C^* e^{-c^* x}. $$

Then, since $|\kappa_p^\alpha(W_k^p)| \le C^*(|W_k^p| + 1)$, the exponential moments for HAR processes lead to

$$ \mathbb{P}(|\kappa_p^\alpha(W_k^p)| \ge x) \le C^* e^{-c^* x}. $$

Thus, since $\mathcal{D}_p \cap [nA, nA + A)$ has less than $(A+1)n_p$ elements, using the above bound with $x \leftarrow x/(A+1)n_p$ lead to

$$ \mathbb{P} \left( \int_{nA}^{(nA+A)-} |\kappa_p^\alpha(W_k^p)| d\mathcal{D}_k^p \ge x \right) \le C^* e^{-c^* x}. $$

Finally, since $\mathrm{Card}(\llbracket -1, \lfloor T/A \rfloor \rrbracket) \times (M \cup [(M \cup P) \times P])) \le C^*T$ it is clear that

$$ \mathbb{P}(Y_T \le x) \ge 1 - C^* T e^{-c^* x}. $$

For the second bound, take $x = \frac{1}{c^*}(q+1) \log(T)$.

Since the proof only uses exponential moments of the process $X = \mathbf{HAR}_{\mathfrak{M}^*}(-\infty)$, it also holds for the process $\mathbf{HAR}_{\mathfrak{M}^*+}(-\infty)$, but since this process dominates the coupling $\mathcal{X}$ which consists in several pieces of processes $\mathbf{HAR}_{\mathfrak{M}^*}(t_0)$ for $t_0 \in \mathbb{R}$, it is clear that the result also holds for $Y_T$. $\square$

**Proof of Lemma E.2.** We follow the steps of [28]. First if $\varphi \in \mathbf{\Phi}^P \cup \mathbf{\Phi}_P$ it is clear that $J_{\varphi,M} = 0$ so the first part is clear. Suppose now that $\varphi \in \mathbf{\Phi}^M \cup \mathbf{\Phi}_M$. From Theorem 3 of [28] applied to the process $J_{\varphi,M}$ with, in the notations of [28], $\mu = 1$, $\varepsilon = 1/2$, $\tau = T$, the following holds.
**Proposition E.4.** Let $\varphi \in \Phi$. Let $x > 0$, $B_{\varphi} > 0$, and $v > w > 0$ constants. Define,

$$V_{\varphi,M} = \frac{1}{3-e} \left( B_{\varphi}^2 x + \sum_{m \in M} \int_0^T \Psi^m(\varphi, t, X_{t-A,t})^2 dS_t^m \right).$$

Then we have,

$$ \mathbb{P} \left( \begin{array}{l} |J_{\varphi,M}| \ge \sqrt{3V_{\varphi,M}}x + \frac{B_{\varphi}x}{3} \quad \text{and} \quad w \le V_{\varphi,M} \le v \quad \text{and} \quad \sup_{m \in M, t \in [0,T]} |\Psi^m(\varphi, t, X_{t-A,t})| \le B_{\varphi} \\ \le 4e^{-x} \left( \frac{\log(v/w)}{\log(3/2)} + 1 \right). \end{array} \right) $$

Let us investigate the choice of $B_{\varphi}$ in order to have lower bounds on the probability of the event

$$ \left( \sup_{m \in M, t \in [0,T]} |\Psi^m(\varphi, t, X_{t-A,t})| \le B_{\varphi} \right). $$

Suppose that $\varphi$ belongs to

1. $\Phi_m$ for $m \in M$, then the choice $B_{\varphi} = 1$ holds with probability 1.

2. $\Phi_{\alpha}^m$ for $m \in M$ and $\alpha \in M \cup P$, then since $\|\Phi\|_{\infty} \le 1$, it is clear that $\Psi^m(\varphi, t, X_{t-A,t}) \le Y_T$ for $0 \le t \le T$. Thus one can take $B_{\varphi} = C^*(q+1)\log(T)$ and it holds with probability at least $1 - \frac{C^*}{T^q}$ by Proposition E.1.

On the other hand, under the event $(\sup_{m \in M, t \in [0,T]} |\Psi^m(\varphi, t, X_{t-A,t})| \le B_{\varphi})$, we have

$$ \frac{B_{\varphi}^2 x}{3 - e} \le V_{\varphi,M} \le \frac{B_{\varphi}^2 (x + S([0,T]))}{3 - e}. $$

From Corollary 5.5, there exist $\theta_0 > 0$ and $K < \infty$ such that $E[e^{\theta_0 S([s,s+1])}] < K$ holds for all $s \in \mathbb{R}$. Thus by Markov inequality we have, for any $0 \le t \le T - 1$,

$$ \mathbb{P} (S([t, t+1]) \ge C^*(q+1) \log(T)) \le \frac{C^*}{T^{q+1}}, $$

and thus by union bound,

$$ \mathbb{P} (S([0, T]) \ge C^*(q+1)T \log(T)) \le \frac{C^*}{T^q}. $$

Thus, with probability at least

$$ 1 - \frac{C^*}{T^q} $$

one can take

$$ w = \frac{B_{\varphi}^2 x}{3 - e}, \quad \text{and} \quad v = \frac{B_{\varphi}^2 (x + C^*(q+1)T \log(T))}{3 - e}, \quad \text{and} \quad B_{\varphi} \text{ the value discussed above.} $$

Combining everything together concludes the proof. $\square$

**Proof of Lemma E.3.** First if $\varphi \in \Phi^M \cup \Phi_M$ it is clear that $J_{\varphi,M} = 0$ so the first part is clear. Suppose now that $\varphi \in \Phi^P \cup \Phi_P$. Recall that the noises $\varepsilon$ are independent and distributed as follows,

$$ \varepsilon_k^p \sim \mathcal{N}(0, \sigma_p^2) $$

with $\max_{p \in P} \sigma_p \le \bar{\sigma}$. The proof is based on Proposition E.5 which can be found in [54] Lemma 4. We prove Proposition E.5 with general notations at the end.
**Proposition E.5.** Let $(H_k^p)_{p \in \mathcal{P}}$ be a predictable process bounded by $B$. Let

$$M_t = \sum_{p \in \mathcal{P}} \int_0^t H_k^p \varepsilon_k^p d\mathcal{D}_k^p$$

and

$$V_t = \sum_{p \in \mathcal{P}} \int_0^t (H_k^p)^2 \sigma_p^2 d\mathcal{D}_k^p.$$

Then we have for all $x \ge 0$,

$$\mathbb{P} \left( |M_t| \ge \sqrt{3x(V_t + B^2\bar{\sigma}^2)} \right) \le 2e^{-x} \left( 1 + \frac{\log \left( 1 + \sum_{p \in \mathcal{P}} (n_p t + 1) \right)}{\log(3/2)} \right).$$

We will apply Proposition E.5 to

$$J_{\varphi, \mathcal{P}} = \sum_{p \in \mathcal{P}} \int_0^T \Psi^p(\varphi, k, X_{k-A,k}) \varepsilon_k^p d\mathcal{D}_k^p.$$

Thus we have to choose wisely $B_\varphi$ to bound with high probability

$$\sup_{(p,k) \in \mathcal{Q}, 0 \le k \le T} |\Psi^p(\varphi, k, X_{k-A,k})|.$$

Suppose that $\varphi$ belongs to

1. $\Phi_p$ for $p \in \mathcal{P}$, then the choice $B_\varphi = 1$ holds with probability 1.

2. $\Phi_\alpha^p$ for $p \in \mathcal{P}$ and $\alpha \in M \cup \mathcal{P}$, then since $\|\Phi\|_\infty \le 1$, it is clear that $\Psi^p(\varphi, k, X_{k-A,k}) \le Y_T$ for $0 \le t \le T$. Thus one can take $B_\varphi = C^*(q+1)\log(T)$ and it holds with probability at least $1 - \frac{C^*}{T^q}$ by Proposition E.1.

This concludes the proof.

□

**Proof of Proposition E.5.** By induction one has that

$$\mathbb{E} \left[ e^{\xi M_t - \frac{\xi^2}{2} V_t} \right] \le 1, \quad \xi \ge 0.$$

Thus for any $x \ge 0$ we have

$$\mathbb{P} \left( M_t \ge \frac{\xi}{2} V_t + \frac{x}{\xi} \right) \le e^{-x}.$$

Then we use a peeling argument. Suppose that $0 \le v_0 \le V_t \le v_1$. Then we have

$$\mathbb{P} \left( M_t \ge \frac{\xi}{2} v_1 + \frac{x}{\xi} \right) \le e^{-x}.$$

Then one can choose the optimal $\xi = \sqrt{2x/v_1}$ and obtains,

$$\mathbb{P} \left( M_t \ge \sqrt{2xv_1} \right) \le e^{-x}.$$

Thus we have

$$\mathbb{P} \left( M_t \ge \sqrt{2xv_1}, v_0 \le V_t \le v_1 \right) \le e^{-x}.$$

Then if $v_1 = (1+\varepsilon)v_0, \dots, v_n = (1+\varepsilon)^n v_0$ we have, by union bound,

$$\mathbb{P} \left( M_t \ge \sqrt{2x(1+\varepsilon)V_t}, v_0 \le V_t \le v_n \right) \le n e^{-x}.$$
We want $v_0 > 0$ thus we apply this result with $V_t + B^2\bar{\sigma} \ge B^2\bar{\sigma} > 0$ instead of $V_t$. Since $(H_k^p)^2 \le B^2$ it is clear that

$$V_t + B^2\bar{\sigma} \le B^2\bar{\sigma}(1 + \sum_{p \in \mathcal{P}} (n_p t + 1)) =: v_n,$$

thus we can take $v_0 = B^2\bar{\sigma}$, $\varepsilon = 1/2$ and $v_n$ as above. Which leads to

$$\mathbb{P}\left(M_t \ge \sqrt{3x(V_t + B^2\bar{\sigma})}\right) \le ne^{-x} \le e^{-x}\left(1 + \frac{\log\left(1 + \sum_{p \in \mathcal{P}} (n_p t + 1)\right)}{\log(3/2)}\right).$$

<figure><img src="image_13.png" /></figure>

## Acknowledgments

I would like to thank my PhD advisors, Vincent Rivoirard and Patricia Reynaud-Bouret for their advices and helpful comments which helped me to improve a lot this article.

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