For the non-stationary process (X', Y'), similarly to the stationary case in (7) on page 3382, define its log-likelihood for all $n \in \mathbb{N}$ and $x \in \mathcal{X}$ as:

$$ \ell'_{n,x}(\theta) := \ell_{n,x}(\theta; Y'_{T_n}), \qquad (79) $$

where $\ell_{n,x}(\theta; \cdot)$ is defined in (6) on page 3382. Moreover, when Assumptions 1-4 and 6 hold and $\Theta$ is compact, similarly to the stationary case in (33) on page 3394, we define the MLE $\hat{\theta}'_{n,x}$ for the non-stationary process as:

$$ \hat{\theta}'_{n,x} = \hat{\theta}'_{n,x}(Y'_{T_n}) \in \operatorname{argmax}_{\theta \in \Theta} \ell'_{n,x}(\theta). \qquad (80) $$

Denote by $\mathbb{P}_{\theta^*, \zeta}$ the probability distribution of the non-stationary HMT process (X', Y'), and by $\mathbb{E}_{\theta^*, \zeta}$ the corresponding expectation.

We can now prove the strong consistency of the MLE for a non-stationary HMT process.

**Theorem 5.1** (Strong consistency of the MLE, non-stationary case). Assume that Assumptions 2-6 hold. the contrast function $\ell$ has a unique maximum (which is then located at $\theta^* \in \Theta$ by Proposition 3.10) and $\Theta$ is compact. Then, the MLE is strongly consistent, that is, for all initial distributions $\zeta$ and all $x \in \mathcal{X}$, the MLE $\hat{\theta}'_{n,x}$ converges $\mathbb{P}_{\theta^*, \zeta}$-a.s. as $n \to \infty$ to the true parameter $\theta^* \in \Theta$.

*Proof.* We start by proving that for any $n \in \mathbb{N}^*$, the distribution of the non-stationary HMT process (X', Y') on $T^*$ is absolutely continuous w.r.t. the distribution of the stationary HMT process (X, Y) on $T^*$, that is:

$$ \mathbb{P}_{\theta^*, \zeta}(X'_{T^*} \in \cdot, Y'_{T^*} \in \cdot) \ll \mathbb{P}_{\theta^*, \pi_{\theta^*}}(X'_{T^*} \in \cdot, Y'_{T^*} \in \cdot) = \mathbb{P}_{\theta^*}(X_{T^*} \in \cdot, Y_{T^*} \in \cdot). \qquad (81) $$

Remind that Assumption 3-(i) implies that $\pi_{\theta^*} \ll \lambda$ with density $\frac{d\pi_{\theta^*}}{d\lambda}$ taking value in $[\sigma^-, \sigma^+]$. Denote by $u_1$ and $u_2$ the two children vertices of $\partial$. Using Assumption 3, for any non-negative measurable function $f$ from $\mathcal{X}^2$ to $\mathbb{R}_+$, we get:

$$ \begin{aligned} & \int_{\mathcal{X}^2} f(x_{u_1}, x_{u_2}) \mathbb{P}_{\theta^*, \zeta}(X'_{u_1} \in dx_{u_1}, X'_{u_2} \in dx_{u_2}) \\ &= \int_{\mathcal{X}^3} f(x_{u_1}, x_{u_2}) q_{\theta^*}(x_{\partial}, x_{u_1}) q_{\theta^*}(x_{\partial}, x_{u_2}) \lambda(dx_{u_1}) \lambda(dx_{u_2}) \zeta(dx_{\partial}) \\ &\leq \left(\frac{\sigma^+}{\sigma^-}\right)^2 \int_{\mathcal{X}^2} f(x_{u_1}, x_{u_2}) \pi_{\theta^*}(dx_{u_1}) \pi_{\theta^*}(dx_{u_2}) \\ &= \left(\frac{\sigma^+}{\sigma^-}\right)^2 \int_{\mathcal{X}^2} f(x_{u_1}, x_{u_2}) \mathbb{P}_{\theta^*}(X_{u_1} \in dx_{u_1}, X_{u_2} \in dx_{u_2}). \end{aligned} $$

In particular, for any measurable subset $A$ of $\mathcal{X}^{T^*} \times \mathcal{Y}^{T^*}$, we can choose $f$ to be defined as:

$$ \begin{aligned} f(x_{u_1}, x_{u_2}) &= \mathbb{E}_{\theta^*, \zeta} [1_A(X'_{T^*}, Y'_{T^*}) | X'_{u_1} = x_{u_1}, X'_{u_2} = x_{u_2}] \\ &= \mathbb{E}_{\theta^*} [1_A(X_{T^*}, Y_{T^*}) | X_{u_1} = x_{u_1}, X_{u_2} = x_{u_2}] \end{aligned} $$

Hence, we get that (81) holds.

Using (81), we get that Proposition 3.7 also holds $\mathbb{P}_{\theta^*, \zeta}$-a.s. with $\ell_{n,x}(\theta)$ replaced by $\ell'_{n,x}(\theta)$, that is, in the non-stationary case. Thus, the proof of Theorem 3.11 can be immediately adapted to the non-stationary case (note that Propositions 3.6 and 3.10 state properties of the contrast function $\ell$, which is the same in the stationary and non-stationary cases). This concludes the proof of the Theorem. $\square$
Using a similar argument as for Theorem 5.1, we get, in the non-stationary case, that the normalized observed information $-|T_n|^{-1}\nabla_{\theta}^2 \ell'_{n,x}(\theta_n)$ converges $P_{\theta^*,\zeta}$-a.s. locally uniformly to the limiting Fisher information $\mathcal{I}(\theta^*)$ (which is defined in (54)). Note that the condition $\rho < 1/2$ on the mixing rate $\rho$ of the HMT process $(X, Y)$ is inherited from Theorem 4.6 in the stationary case. See the discussion in Remark 1.5 for comments on this condition on $\rho$.

**Theorem 5.2** (Convergence of the normalized observed information, non-stationary case). Assume that Assumptions 2–4 and 6–9 hold. Assume that $\rho < 1/2$. Assume that $\Theta$ is compact. Then, for all initial distributions $\zeta$ and all $x \in \mathcal{X}$, we have:

$$ \lim_{\delta \to 0} \lim_{n \to \infty} \sup_{\theta \in \mathcal{O}: \|\theta - \theta^*\| \le \delta} \left\| -|T_n|^{-1} \nabla_{\theta}^2 \ell'_{n,x}(\theta) - \mathcal{I}(\theta^*) \right\| = 0 \quad \mathbb{P}_{\theta^*, \zeta}\text{-a.s.} $$

In particular, combining Theorems 5.1 and 5.2, we get that the normalized observed information $-|T_n|^{-1}\nabla_{\theta}^2 \ell_{n,x}(\hat{\theta}_{n,x})$ at the MLE $\hat{\theta}_{n,x}$ is a strongly consistent estimator of the Fisher information matrix $\mathcal{I}(\theta^*)$.

Before proving the asymptotic normality of the MLE in the non-stationary case, we start with the following lemma which present a coupling construction for the two processes $(X, Y)$ and $(X', Y')$.

**Lemma 5.3** (Coupling construction of two HMTs). Assume that Assumptions 2 and 3 hold. Further assume that $\sigma^- \ge 1/2$. Then, it is possible to construct the two processes $(X, Y)$ and $(X', Y')$ on a common probability space such that there exists an a.s. finite random time $N$, which we call the coupling time, such that $(X_u, Y_u) = (X'_u, Y'_u)$ for all $u \in T_n$ with $n \ge N$.

We will denote by $P_{\theta^* \succsim \zeta}$ the probability distribution that realizes this coupling.

Note that $\rho \le 1/2$ implies that $\sigma^- \ge \sigma^+/2 \ge 1/2$ (see Assumption 3).

*Proof.* We first construct the coupling only for the process $X$ and $X'$. We define the coupling construction inductively on the height of the tree. For the root vertex, we use an independent coupling construction for $X_\partial$ and $X'_\partial$, which are distributed according to $\pi_{\theta^*}$ and $\zeta$ respectively (note that it is also possible to use a perfect coupling with probability error $\|\pi_{\theta^*} - \zeta\|_{TV}$). Then, if the coupling has been constructed up to generation $n \in \mathbb{N}$, using the Markov property, we proceed to construct independently the coupling for each vertices in generation $n+1$. Let $u \in G_{n+1}$. If the variables were already coupled for the parent vertex $p(u)$, that is $X_{p(u)} = X'_{p(u)}$, then we choose the new value $X_u = X'_u$ according to the transition kernel $Q_{\theta^*}$. Otherwise, if $X_{p(u)} \ne X'_{p(u)}$, then using the uniform geometric ergodicity (remind Assumption 3) of the transition kernel $Q_{\theta^*}$, we know that $\sup_{X,x' \in \mathcal{X}} \|Q_{\theta^*}(x; \cdot) - Q_{\theta^*}(x'; \cdot)\|_{TV} \le 1 - \sigma^-$, and thus we can construct a coupling of $X_u$ and $X'_u$ conditionally on $X_{p(u)} \ne X'_{p(u)}$ with exact matching probability at least $1 - \sigma^-$. We have constructing the matching for $u$, and thus for the whole generation $n+1$. Using Kolmogorov's extension theorem, there exists a coupling measure for the whole tree $T$ whose finite dimensional marginals are the ones given above. Recall that Kolmogorov's extension theorem holds for general Polish spaces and not just the real line (see (Bogachev, 2007, Theorem 7.7.1 with Theorem 7.1.7)), and also recall that $\mathcal{X}$ and $\mathcal{Y}$, and thus also $\mathcal{X} \times \mathcal{Y}$, are Polish spaces.

Remark that the joint process $(X, Y)$ satisfies a uniform geometric ergodicity bound with the same constant $1 - \sigma^-$. Thus, the construction above can be extended to the joint process $(X, Y)$. Denote by $P_{\theta^* \succsim \zeta}$ the probability distribution of the coupling we have constructed for the joint process $(X, Y)$.
Define the random coupling time $N = \inf\{n \in \mathbb{N} | \forall u \in G_n, X_u = X'_u\}$, which is the first generation for which the exact coupling occurs for all vertices (and $N = \infty$ is this never happens). We are left to prove that $\mathbb{P}_{\theta^{\star \triangleright \alpha \zeta}}$-a.s. $N < \infty$. We say that a vertex $u$ is a special vertex if $X_{u} \neq X'_{u}$. Note that if $u$ is not special, then all its descendants are also not special. Also note that special vertices form a Bienaymé-Galton-Watson tree whose (homogeneous) offspring distribution takes the values: 0 with probability $(\sigma^{-})^{2}$; 1 with probability $2\sigma^{-}(1 - \sigma^{-})$; and 2 with probability $(1 - \sigma^{-})^{2}$. The average of this offspring distribution is $2(1 - \sigma^{-})$. Hence, the number of special vertices if finite $\mathbb{P}_{\theta^{\star \triangleright \alpha \zeta}}$-a.s., that is, $N$ is finite $\mathbb{P}_{\theta^{\star \triangleright \alpha \zeta}}$-a.s., if and only if $2(1 - \sigma^{-}) \le 1$, that is, $\sigma^{-} \ge 1/2$. This concludes the proof. $\square$

Remind that the log-likelihood function $\ell_{n,x}$ (resp. $\ell'_{n,x}$), which is a random function depending on $Y_{T_n}$ from the stationary HMT process (resp. on $Y'_{T_n}$ from the non-stationary HMT process), is defined in (7) on page 3382 (resp. just before (79) on page 3425). For all $\theta \in \Theta$, define:

$$D_{n,x}(\theta) = \ell'_{n,x}(\theta) - \ell_{n,x}(\theta) \\ = \sum_{u \in T_n} \log p_{\theta}(Y'_{u} | Y^{\Delta^{*}(u,h(u))}, X'_{\theta} = x) - \log p_{\theta}(Y_{u} | Y^{\Delta^{*}(u,h(u))}, X_{\theta} = x),$$

where remind that $p_{\theta}$ denotes possibly conditional density (see (5) on page 3382).

Remind that when Assumptions 1–4 and 6 hold and $\Theta$ is compact, for all $x \in \mathcal{X}$, the MLE $\hat{\theta}_{n,x}$ (resp. $\hat{\theta}'_{n,x}$) is a random variable which depends on $Y_{T_n}$ from the stationary HMT process (resp. on $Y'_{T_n}$ from the non-stationary HMT process) and is defined in (33) on page 3394 (resp. in (80) on page 3425).

To prove that $\lim_{n \to \infty} |T_n|^{1/2}(\hat{\theta}'_{n,x} - \hat{\theta}_{n,x}) = 0$ $\mathbb{P}_{\theta^{\star \triangleright \alpha \zeta}}$-a.s., and thus that $\hat{\theta}'_{n,x}$ and $\hat{\theta}_{n,x}$ are asymptotically normal with the same covariance matrix (remind Theorem 4.7), we must first prove that the function $\theta \mapsto D_{n,x}(\theta)$ satisfies some kind of continuity property. Note that we proved in the proof of Theorem 5.1 that Proposition 3.7 holds both in the stationary and the non-stationary cases, and thus we have:

$$\lim_{n \to \infty} \sup_{\theta \in \Theta} |T_n|^{-1} D_{n,x}(\theta) = 0 \quad \mathbb{P}_{\theta^{\star \triangleright \alpha \zeta}} \text{-a.s.}$$

However, we need some kind of continuity property without the normalizing term $||T_n|^{-1}$, which is given by the following lemma.

**Lemma 5.4.** Assume that Assumptions 2–6 hold. Further assume that $\rho < 1/2$. Then, for all initial distributions $\zeta$ and all $x \in \mathcal{X}$, we have:

$$\lim_{n \to \infty} |D_{n,x}(\hat{\theta}'_{n,x}) - D_{n,x}(\hat{\theta}_{n,x})| = 0, \quad \mathbb{P}_{\theta^{\star \triangleright \alpha \zeta}} \text{-a.s.}$$

*Proof.* [The proof of the lemma is a straightforward adaptation of the proof of (Douc, Moulines and Rydén, 2004, Lemmas 11 and 12).]

Let $N$ be the random time provided by Lemma 5.3. We first prove that $\mathbb{P}_{\theta^{\star \triangleright \alpha \zeta}}$-a.s., we have:

$$\sum_{u \in T \setminus T_N} \sup_{\theta \in \Theta} |\log p_{\theta}(Y'_{u} | Y'_{\Delta^{*}(u,h(u))}, X'_{\theta} = x) - \log p_{\theta}(Y_{u} | Y^{\Delta^{*}(u,h(u))}, X_{\theta} = x)| < \infty. \quad (82)$$
Note that for all $u \in T_n$ and $v$ an ancestor of $u$ (distinct of $u$), we have:

$$p_{\theta}(Y_u | Y_{\Delta^*(u,h(u))}), X_{\partial} = x) = \int_{X^3} g_{\theta}(x_u, Y_u) q_{\theta}(x_{p(u)}, x_u) \lambda(dx_u) \\ \times \mathbb{P}_{\theta}(X_{p(u)} \in dx_{p(u)} | X_v = x_v, Y_{\Delta^*(u,h(u)-h(v))}) \\ \times \mathbb{P}_{\theta}(X_v \in dx_v | Y_{\Delta^*(u,h(u))}, X_{\partial} = x),$$

and similarly for $p_{\theta}(Y'_u | Y'_{\Delta^*(u,h(u))}), X'_{\partial} = x)$. Using the fact that for $v \in T$ with height $h(v) \ge N$, we have $Y_v = Y'_v$, and using Lemma 3.2, we have for all $u \in T$ with height $h(u) > N$:

$$|p_{\theta}(Y'_{u} | Y'_{\Delta^{*}(u,h(u))}, X'_{\partial} = x) - p_{\theta}(Y_{u} | Y_{\Delta^{*}(u,h(u))}, X_{\partial} = x)| \\ \le 2 \rho^{h(u)-N-1} \sigma^{+} \int g_{\theta}(x, Y_{u}) \lambda(dx).$$

Thus, using a similar argument as in the proof of Lemma 3.3, we get:

$$| \log p_{\theta}(Y'_{u} | Y'_{\Delta^{*}(u,h(u))}, X'_{\partial} = x) - \log p_{\theta}(Y_{u} | Y_{\Delta^{*}(u,h(u))}, X_{\partial} = x) | \le \frac{\rho^{h(u)-N-1}}{1-\rho}.$$

Hence, the sum in (82) is $\mathbb{P}_{\theta^{\star\Box\Box\zeta}}$-a.s. upper bounded by a constant times $\sum_{k=N+1}^{\infty} 2^k \rho^k \le (2\rho)^{N+1}/(1-2\rho)$, and is thus finite $\mathbb{P}_{\theta^{\star\Box\Box\zeta}}$-a.s.

Let $\varepsilon > 0$. Using (82), there exists a random integer $N_{\varepsilon}$ which $\mathbb{P}_{\theta^{\star\Box\Box\zeta}}$-a.s. is finite and satisfies (remind that $\Delta^*(u) = \Delta^*(u, h(u))$):

$$\sum_{u \in T \setminus T_{N_{\varepsilon}}} \sup_{\theta \in \Theta} | \log p_{\theta}(Y'_{u} | Y'_{\Delta^{*}(u)}, X'_{\partial} = x) - \log p_{\theta}(Y_{u} | Y_{\Delta^{*}(u)}, X_{\partial} = x) | \le \varepsilon.$$

Thus, $\mathbb{P}_{\theta^{\star\Box\Box\zeta}}$-a.s., for all $n \ge N_{\varepsilon}$, we have:

$$|D_{n,x}(\hat{\theta}'_{n,x}) - D_{n,x}(\hat{\theta}_{n,x})| \le 2\varepsilon + |\ell'_{N_{\varepsilon},x}(\hat{\theta}'_{n,x}) - \ell'_{N_{\varepsilon},x}(\hat{\theta}_{n,x})| \\ + |\ell_{N_{\varepsilon},x}(\hat{\theta}'_{n,x}) - \ell_{N_{\varepsilon},x}(\hat{\theta}_{n,x})|.$$

Note that under the given assumptions, the functions $\theta \mapsto \ell'_{N_{\varepsilon},x}(\theta)$ and $\theta \mapsto \ell_{N_{\varepsilon},x}(\theta)$ are continuous $\mathbb{P}_{\theta^{\star\Box\Box\zeta}}$-a.s. (see the proof of Proposition 3.6). Hence, the proof is complete upon observing that $\hat{\theta}'_{n,x}$ and $\hat{\theta}_{n,x}$ both converge $\mathbb{P}_{\theta^{\star\Box\Box\zeta}}$-a.s. to $\theta^{\star}$ (see Theorem 5.1), and that $\varepsilon$ was arbitrary. $\square$

We can now prove the asymptotic normality of the MLE $\hat{\theta}'_{n,x}$ in the non-stationary case. Remind that the contrast function $\ell$ is defined in (26) on page 3390.

**Theorem 5.5** (Asymptotic normality of the MLE, non-stationary case). Assume that Assumptions 2–9 hold. Assume that $\rho < 1/2$. Further assume that the contrast function $\ell$ has a unique maximum (which is then located at $\theta^{\star} \in \Theta$ by Proposition 3.10) and that $\Theta$ is compact, $\theta^{\star}$ is an interior point of $\Theta$, and the limiting Fisher information matrix $\mathcal{I}(\theta^{\star})$ (which is defined in (54)) is non-singular. Then, for all initial distributions $\zeta$ and for all $x \in \mathcal{X}$, we have:

$$|T_n|^{1/2} (\hat{\theta}'_{n,x} - \theta^{\star}) \xrightarrow{n \to \infty} \mathcal{N}(0, \mathcal{I}(\theta^{\star})^{-1}) \quad \text{under } \mathbb{P}_{\theta^{\star}, \zeta},$$

where $\mathcal{N}(0, M)$ denotes the centered Gaussian distribution with covariance matrix $M$.
**Proof.** [The proof of the theorem is a straightforward adaptation of the proof of (Douc, Moulines and Rydén, 2004, Theorem 6).]

Define $\varepsilon_n = |T_n|^{1/2}(\hat{\theta}_{n,x} - \hat{\theta}'_{n,x})$ for all $n \in \mathbb{N}$, and remark that it is sufficient to prove that $\lim_{n \to \infty} \varepsilon_n = 0 \mathbb{P}_{\theta^{* \uparrow \downarrow \zeta}}$-a.s. Since $\hat{\theta}'_{n,x}$ is the maximizer of the function $\theta \mapsto \ell'_{n,x}(\theta)$, we have that $\ell'_{n,x}(\hat{\theta}'_{n,x}) \ge \ell'_{n,x}(\hat{\theta}_{n,x})$. Thus, using a Taylor expansion of $\ell_{n,x}$ around its maximizer $\hat{\theta}_{n,x}$ (for which we have $\nabla_{\theta}\ell_{n,x}(\hat{\theta}_{n,x}) = 0$), we get that there exists $t_n \in [0, 1]$ such that:

$$D_{n,x}(\hat{\theta}'_{n,x}) - D_{n,x}(\hat{\theta}_{n,x}) \ge \ell_{n,x}(\hat{\theta}_{n,x}) - \ell_{n,x}(\hat{\theta}'_{n,x}) \\ = -\frac{1}{2}|T_n|^{-1}\varepsilon_n' \nabla_{\theta}^2 \ell_{n,x}(t_n \hat{\theta}'_{n,x} + (1-t_n)\hat{\theta}_{n,x})\varepsilon_n.$$

Note that we have $\lim_{n \to \infty} t_n \hat{\theta}'_{n,x} + (1 - t_n)\hat{\theta}_{n,x} = \theta^* \mathbb{P}_{\theta^{* \uparrow \downarrow \zeta}}$-a.s. by Theorem 5.1. Thus, applying Theorem 4.6, we have:

$$\lim_{n \to \infty} -|T_n|^{-1} \nabla_{\theta}^2 \ell_{n,x}(t_n \hat{\theta}'_{n,x} + (1-t_n)\hat{\theta}_{n,x}) = \mathcal{I}(\theta^*), \quad \mathbb{P}_{\theta^{* \uparrow \downarrow \zeta}} \text{-a.s.}$$

As $\mathcal{I}(\theta^*)$ is positive definite, there exits $M > 0$ such that on a set with $\mathbb{P}_{\theta^{* \uparrow \downarrow \zeta}}$-probability one and for $n$ sufficiently large, we have:

$$D_{n,x}(\hat{\theta}'_{n,x}) - D_{n,x}(\hat{\theta}_{n,x}) \ge M|\varepsilon_n|^2.$$

Then, the proof is complete by applying Lemma 5.4. □

## Appendix A: Ergodic theorem for Markov processes indexed by trees with neighborhood-dependent functions

In this appendix, we prove generalization of the ergodic theorems in Guyon (2007) and in Weibel (2025), which give a.s. and $L^2$ convergences for branching Markov chains, to allow for neighborhood-dependent functions. Those ergodic theorems are used to prove the ergodic convergence lemmas in Section 2.4 which are used in the main body of this article. Remind that we need those generalization as in the study of asymptotic property of the MLE for the HMT relies on the study of the likelihood contribution functions $h_{u,k,x}(\theta; Y_{\Delta(u,k)})$ (defined in (17) on page 3387) which are neighborhood-dependent.

Remind from Section 2 that if $(X, Y)$ is a HMT process, then the joint process $((X_u, Y_u), u \in T)$ is a branching Markov chain. Thus, it is enough to prove those ergodic theorems for branching Markov chains instead of HMT processes.

Let $Q$ be a transition kernel on $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$ where $\mathcal{X}$ is a metric space. We assume that $Q$ has a unique invariant probability distribution $\pi$ and is uniformly geometrically ergodic, that is, there exists $\rho \in (0, 1)$ and a finite positive constant $C$ such that for all $x \in \mathcal{X}$, we have $\|Q^n(x; \cdot) - \pi\|_{TV} \le C\rho^n$. Remind from Lemma 2.3 that this covers the case $Q = Q_{\theta}$ for any $\theta \in \Theta$. Let $X = (X_u, u \in T)$ be a branching Markov chain with transition kernel $Q$ and initial distribution $\pi$. Denote by $\mathbb{P}_Q$ the probability distribution of the process $X$, and by $\mathbb{E}_Q$ the corresponding expectation.

In this section, for a probability measure $\nu$ on $\mathcal{X}$, a transition kernel $Q$ on $(\mathcal{X}, \mathcal{B}(\mathcal{Z}))$ and a Borel integrable function $f$ on $\mathcal{B}(\mathcal{Z})$ where $\mathcal{Z} = \mathcal{X}^A$ for some finite subset $A \subset T$, we will write $\nu Q$ for the image probability measure $(\nu Q)(\cdot) = \int_{\mathcal{X}} Q(x; \cdot) \nu(dx)$, and $Qf$ for the Borel function $(Qf)(x) = \int_{\mathcal{Z}} f(z) Q(x; dz)$. For a probability measure $\nu$ on $\mathcal{X}$ and a Borel integrable function $f$ on $\mathcal{X}$, we will write $\langle \nu, f \rangle = \nu f = \int_{\mathcal{X}} f \, dv$.
We will need the following lemma which states geometric convergence bounds for functions in $L^2(\pi)$.

**Lemma A.1** (Convergence bounds when $Q$ is uniformly geometrically ergodic). Assume that the transition kernel $Q$ has a unique invariant measure $\pi$, and that $Q$ is uniformly geometrically ergodic. Then, there exists finite positive constants $\alpha \in (0, 1)$ and $M < \infty$ such that for all functions $f \in L^2(\pi)$, we have:

$$ \forall n \in \mathbb{N}, \quad \sup_{k \in \mathbb{N}} \pi Q^k (Q^n f - \langle \pi, f \rangle)^2 = \pi(Q^n f - \langle \pi, f \rangle)^2 \le M \alpha^{2n} \|f - \langle \pi, f \rangle\|_{L^2(\pi)}^2. $$

In particular, the function $f$ satisfies $\sup_{n \in \mathbb{N}} \pi Q^n f^2 < \infty$.

Note, using Cauchy-Schwarz and Jensen's inequalities, that $\sup_{n \in \mathbb{N}} \pi Q^n f^2 < \infty$ implies that $Q^n f, Q^n f^2$, and $Q^k(Q^n f \times Q^m f)$ (with $n, m, k \in \mathbb{N}$) are well-defined and finite $\pi$-almost everywhere and are $\pi$-integrable.

*Proof.* Using (Douc et al., 2018, Proposition 22.3.5 and Definition 22.3.1), we get that there exists finite positive constants $\alpha \in (0, 1)$ and $M < \infty$ such that for all functions $f \in L^2(\pi)$, we have $\pi(Q^n f - \langle \pi, f \rangle)^2 = \|Q^n(f - \langle \pi, f \rangle)\|_{L^2(\pi)}^2 \le M \alpha^{2n} \|f - \langle \pi, f \rangle\|_{L^2(\pi)}^2$ for all $n \in \mathbb{N}$. In particular, we get that $\sup_{n \in \mathbb{N}} \pi Q^n f^2 < \infty$. $\square$

Let $k \in \mathbb{N}$ be fixed. Recall from Section 2.4 the definitions of the subtrees $\Delta(u, k)$, of their shapes $\mathcal{S}h(\Delta(u, k))$ (defined in (8)), and of the finite set of possible shapes $\mathcal{N}_k$ (defined in (9)). For simplicity, in this appendix we will write $\mathcal{S}_u$ instead of $\mathcal{S}h(\Delta(u, k))$. Also remind from Section 2.4 the definition of a collection of neighborhood-shape-dependent functions ($f_S : \mathcal{X}^S \to \mathbb{R})_{S \in \mathcal{N}_k}$. Remind that for such a collection of functions, we simply write $f_{\Delta(u,k)}$ or $f_{S_u}$ instead of $f_{\mathcal{S}h(\Delta(u,k))}$. And also remind that we write $f_{S_u}(X_{\Delta(u,k)})$ for the evaluation of $f_{S_u} = f_{\Delta(u,k)}$ on $X_{\Delta(u,k)}$. Note that up to translation, we may identify $\mathcal{X}^S$ and $\mathcal{X}^{\Delta(u,k)}$ for any $u \in T \setminus T_{k-1}$ such that $\mathcal{S}_u = \mathcal{S}$.

Remind that any subset $A \subset T$, we denote by $X_A$ the gathered variables ($X_v : v \in A$). For a collection of neighborhood-shape-dependent functions $f = (f_S : \mathcal{X}^S \to \mathbb{R})_{S \in \mathcal{N}_k}$, define the empirical average of $f$ over a finite subset $A \subset T \setminus T_{k-1}$ as:

$$ \bar{M}_A(f) = |A|^{-1} \sum_{u \in A} f_{S_u}(X_{\Delta(u,k)}). \qquad (83) $$

For a neighborhood shape $S \in \mathcal{N}_k$, let $u \in G_k$ be the unique vertex such that $S_u = S$, and define the transition kernel $Q^S$ on $(\mathcal{X}, \mathcal{B}(\mathcal{X}^S))$ for any $x \in \mathcal{X}$ and any Borel function $f$ on $\mathcal{X}^S = \mathcal{X}^{\Delta(u)}$ which is in $L^1(X_{\Delta(u)}) = L^1(X_S)$ by:

$$ Q^S f(x) = \mathbb{E}_Q \left[ f(X_{\Delta(u)}) \mid X_{\partial} = x \right]. \qquad (84) $$

That is, from the value $x \in \mathcal{X}$ of the root vertex $v$ in $S$, the transition kernel $Q^S$ returns the distribution of the Markov process $X$ on $S$ with transition kernel $Q$ conditioned on the value $X_v$ of the vertex $v$ being $x$. Note that (84) also extends to any vertex $u \in T \setminus T_{k-1}$ such that $S_u = S$, which gives us:

$$ Q^{S_u} f(x) = \mathbb{E}_Q \left[ f(X_{\Delta(u,k)}) \mid X_{p^k(u)} = x \right]. \qquad (85) $$

Moreover, using Jensen's inequality, note that if $f \in L^2(X_S)$, then $Q^S f$ is in $L^2(\pi) = L^2(X_{\partial})$.
Remind that as $T$ is a plane rooted tree, we can enumerate its vertices as a sequence $(v_j)_{j \in \mathbb{N}}$ in a breadth-first-search manner, that is, which is increasing for $<$ (note that $u_0 = \partial$). Also remind that, for $n \ge |T_{k-1}|$, if $V_n$ is uniformly distributed over $A_n := \{v_j : |T_{k-1}| < j \le n\} = \Delta(v_n) \setminus T_{k-1}$, then the distribution of $S_{V_n}$ converges to the uniform distribution over $\mathcal{N}_k$ as $n \to \infty$.

We are now ready to state the ergodic theorem with neighborhood-shape-dependent functions for branching Markov chains indexed by the infinite complete binary tree $T$. Remind that $\bar{M}_{A_n}(f)$ is defined in (83).

**Theorem A.2 (Ergodic theorem with neighborhood-dependent functions).** Let $k \in \mathbb{N}$ be fixed. Let $(v_j)_{j \in \mathbb{N}}$ be the sequence enumerating the vertices in $T$ in a breadth-first-search manner. For all $n > |T_{k-1}|$, define $A_n = \Delta(v_n) \setminus T_{k-1}$.

Let $Q$ be a transition kernel on $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$ which is uniformly geometrically ergodic and has a unique invariant probability measure $\pi$. Let $X = (X_u, u \in T)$ be a branching Markov chain with transition kernel $Q$ and initial distribution $\pi$.

Let $f = (f_S : \mathcal{X}^S \to \mathbb{R})_{S \in \mathcal{N}_k}$ be a collection of neighborhood-shape-dependent Borel functions that are in $L^2(X)$. Then, we have:

$$ \bar{M}_{A_n}(f) \xrightarrow[n \to \infty]{\phantom{=} \mathbb{E}_{U_k} \otimes \mathbb{E}_Q} \mathbb{E}_{U_k} \left[ f_{S_{U_k}}(X_{\Delta(U_k)}) \right] \quad \text{in } L^2(\pi) = L^2(X), \quad (86) $$

where $U_k$ is uniformly distributed over $G_k$ and independent of the process $X$, and $\mathbb{E}_{U_k} \otimes \mathbb{E}_Q$ denotes the joint expectation over $U_k$ and $X$.

As an immediate corollary, we get that the result still holds if $A_n$ is replaced by $T_n \setminus T_{k-1}$.

**Remark A.3 (More general assumptions).** Note that without changing the proof, we could replace the subtrees $\Delta(u, k)$ by general subtrees (i.e. a connected subsets) $O_u$ of the $k$-neighborhood $B_T(u, k) := \{v \in T : d(u, v) \le k\}$ of $u$ such that $O_u$ contains $u$. In that case, we must assume that the distribution of the shape (i.e. when seen up to translation) $\mathcal{S}h(O_{V_n})$ converges to some limit distribution. Also note that we could allow more general choices as in Weibel (2025) for the averaging sets $(A_n)_{n \in \mathbb{N}}$, for the tree $T$, and for the transition kernel $Q$ and initial distribution of the branching Markov chain $X$.

*Proof. First case: we only have constant functions* $f_S \equiv c(S)$ **for all $S \in \mathcal{N}_k$**. Then, for every $n \in \mathbb{N}$ we have:

$$ \bar{M}_{A_n}(f) = \sum_{S \in \mathcal{N}_k} c(S) \frac{|u \in A_n : S_u = S|}{|A_n|}, \quad (87) $$

where the right hand side converges in distribution (and thus in $L^2$) as $n \to \infty$ to $\mathbb{E}_{U_k}[c(S_{U_k})]$ (remind that the distribution of $S_{V_n}$ converges to the uniform distribution on $G_k$ when $n \to \infty$). This concludes the proof in this first case.

**General case:** We adapt the proof of (Weibel, 2025, Theorem 2.2) where the single function is replaced by a family of neighborhood-shape-dependent functions.

Using the first case and the linearity in $f$ of the empirical averages, and replacing $f_S$ by $f_S - \langle \pi, \mathcal{Q}^S f_S \rangle$, we may assume that $\langle \pi, \mathcal{Q}^S f_S \rangle = 0$ for all $S \in \mathcal{N}_k$. For all $n \in \mathbb{N}$, we have:

$$ \mathbb{E}_Q \left[ \bar{M}_{A_n}(f)^2 \right] = \frac{1}{|A_n|^2} \sum_{u,v \in A_n} \mathbb{E}_Q \left[ f_{S_u}(X_{\Delta(u,k)}) f_{S_v}(X_{\Delta(v,k)}) \right]. \quad (88) $$
Using Lemma A.1, as the transition kernel $Q$ is uniformly geometrically ergodic and has unique invariant probability measure $\pi$, and as the function $Q^S f_S$ for $S \in \mathcal{N}_k$ are all in $L^2(\pi)$ (see the comment just after (85)), we have that $C_S := \sup_{n \in \mathbb{N}} \pi Q^n(Q^S f_S)^2 < \infty$ for all $S \in \mathcal{N}_k$. Define $C := \max_{S \in \mathcal{N}_k} C_S < \infty$ (remind that $\mathcal{N}_k$ is finite). Thus, for $u \in T$, we have:

$$ \mathbb{E}_Q \left[ f_{S_u} (X_{\Delta(u,k)})^2 \right] = \pi Q^{(h(u)-k)_+} (Q^{S_u} f_{S_u})^2 \le C < \infty. $$

Hence, for all $u, v \in T$, using Cauchy-Schwarz inequality, we have:

$$ \begin{aligned} \mathbb{E}_Q \left[ f_{S_u}(X_{\Delta(u,k)}) f_{S_v}(X_{\Delta(v,k)}) \right] & \le \left( \mathbb{E}_Q \left[ f_{S_u}(X_{\Delta(u,k)})^2 \right] \mathbb{E}_Q \left[ f_{S_v}(X_{\Delta(v,k)})^2 \right] \right)^{1/2} \\ & \le C < \infty. \end{aligned} \quad (89) $$

Let $u, v \in T$ such that $d(u, v) > 2k$, which implies that $\Delta(u, k) \cap \Delta(v, k) = \emptyset$. Without loss of generality, assume that $h(u) \ge h(v)$. Then, we have that $h(u \wedge v) < h(u) - k$. Denote by $v_0$ the last ancestor of $u$ in $\Delta(v, k) \cup \{u \wedge v\}$. Remark that $u \wedge v$ is an ancestor of $v_0$. Then, we have:

$$ \begin{aligned} & \mathbb{E}_Q \left[ f_{S_u}(X_{\Delta(u,k)}) f_{S_v}(X_{\Delta(v,k)}) \right] \\ &= \mathbb{E}_Q \left[ \mathbb{E}_Q \left[ f_{S_u}(X_{\Delta(u,k)}) \mid X_{\Delta(v,k)}, X_{u \wedge v} \right] f_{S_v}(X_{\Delta(v,k)}) \right] \\ &\le \left( \mathbb{E}_Q \left[ \mathbb{E}_Q \left[ f_{S_u}(X_{\Delta(u,k)}) \mid X_{\Delta(v,k)}, X_{u \wedge v} \right]^2 \right] \mathbb{E}_Q \left[ f_{S_v}(X_{\Delta(v,k)})^2 \right] \right)^{1/2} \\ &\le C^{1/2} \left( \mathbb{E}_Q \left[ \mathbb{E}_Q \left[ f_{S_u}(X_{\Delta(u,k)}) \mid X_{v_0} \right]^2 \right] \right)^{1/2} \\ &\le C^{1/2} \left( \mathbb{E}_Q \left[ \mathbb{E}_Q \left[ f_{S_u}(X_{\Delta(u,k)}) \mid X_{u \wedge v} \right]^2 \right] \right)^{1/2} \\ &= C^{1/2} \left( \pi Q^{h(u \wedge v)} (Q^{d(u \wedge v, u)} - k) Q^{S_u} f_{S_u} \right)^{1/2} \\ &\le C^{1/2} \left( \max_{S \in \mathcal{N}_k} \pi Q^{h(u \wedge v)} (Q^{\tilde{d}(u,v)} - k) Q^S f_S \right)^{1/2}, \end{aligned} \quad (90) $$

where we used Cauchy-Schwarz inequality in the first inequality, we used (89) and the Markov property of the process $X$ in the second inequality, and we used Jensen's inequality in the third inequality. Remark that $\tilde{d}(u, v) = \max(d(u \wedge v, u), d(u \wedge v, v))$ is a distance on $T$ that satisfies $d/2 \le \tilde{d} \le d$.

Let $U_n$ and $V_n$ be uniformly distributed over $A_n$, and independent of each other and of the branching Markov chain $X$, and denote by $\mathbb{P}_{U_n, V_n}$ their joint probability distribution. Using again Lemma A.1, there exist finite constants $M < \infty$ and $\alpha \in (0, 1)$ such that for all $S \in \mathcal{N}$ we have

$$ \forall m \in \mathbb{N}, \quad \sup_{j \in \mathbb{N}} \pi Q^j (Q^m Q^S f_S)^2 \le M^2 \alpha^{2m}. \quad (91) $$

Hence, combining (88), (89), (90) and (91), we get for any $K \ge k$:

$$ \mathbb{E}_Q \left[ \bar{M}_{A_n} (f)^2 \right] $$
$$
\begin{align}
&\le C \mathbb{P}_{U_n, V_n} (\tilde{d}(U_n, V_n) \le 2K) \nonumber \\
&\quad + C^{1/2} |A_n|^{-2} \sum_{u,v \in A_n: \tilde{d}(u,v) > 2K} \left( \max_{S \in \mathcal{N}_k} \pi Q^{h(u \wedge v)} (Q^{\tilde{d}(u,v)-K} Q^S f_S)^2 \right)^{1/2} \nonumber \\
&\le C \mathbb{P}_{U_n, V_n} (d(U_n, V_n) \le 4K) \nonumber \\
&\quad + C^{1/2} |A_n|^{-2} \sum_{u,v \in A_n: \tilde{d}(u,v) > 2K} M \alpha^{\tilde{d}(u,v)-K} \tag{92} \\
&\le C \mathbb{P}_{U_n, V_n} (d(U_n, V_n) \le 4K) + C^{1/2} M \alpha^K. \tag{93}
\end{align}
$$

Let $\varepsilon > 0$. Let $K \ge k$ be such that $C^{1/2}M\alpha^K < \varepsilon$. Using (Weibel, 2025, Lemma 3.1), we get that the first term in the right hand side of (93) goes to zero as $n \to \infty$. Thus, for $n$ large enough, the right hand side of (93) is upper bounded by $2\varepsilon$. This being true for all $\varepsilon > 0$, we get that $\lim_{n\to\infty} \mathbb{E}_Q[\bar{M}_{A_n}(f)^2] = 0$. This concludes the proof. $\square$

We now state and prove a strong law of large numbers for branching Markov chains indexed
by the infinite complete binary tree T and with neighborhood-shape-dependent functions.
This result uses the same assumptions as in Theorem A.2.

**Theorem A.4** (Strong law of larger numbers with neighborhood-dependent function). Let $Q$ be a transition kernel on $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$ which is uniformly geometrically ergodic and has a unique invariant probability measure $\pi$. Let $X = (X_u, u \in T)$ be a branching Markov chain with transition kernel $Q$ and initial distribution $\pi$.

Let $k \in \mathbb{N}$ be fixed. Let $U_k$ be uniformly distributed over $G_k$ and independent of the process $X$, and let $\mathbb{E}_{U_k} \otimes \mathbb{E}_Q$ denote the joint expectation over $U_k$ and $X$. Let $f = (f_S : X^S \to \mathbb{R})_{S \in \mathcal{N}_k}$ be a collection of neighborhood-shape-dependent Borel functions that are in $L^2(X)$.

Then, we have:

$$
\text{a.s.} \quad \lim_{n \to \infty} \bar{M}_{G_n}(f) = \lim_{n \to \infty} \bar{M}_{T_n \setminus T_{k-1}}(f) = \mathbb{E}_{U_k} \otimes \mathbb{E}_Q [f_{S_{U_k}}(X_{\Delta(U_k)})].
$$

Moreover, there exist finite constants $C_0 < \infty$ and $\beta \in (0, 1)$ such that:

$$
\forall n \ge k, \quad \mathbb{E}_Q \left[ \left( \bar{M}_{G_n}(f) - \mathbb{E}_{U_k} \otimes \mathbb{E}_Q [f_{S_{U_k}}(X_{\Delta(U_k)})] \right)^2 \right] \le C_0 \beta^n. \quad (94)
$$

*Proof.* After using (92), the proof is an easy adaptation of the proof of (Guyon, 2007, Theorem 14).

The case of $\bar{M}_{T_n \setminus T_{k-1}}(f)$ follows directly from the case of $\bar{M}_{G_n}(f)$ as:

$$
|\bar{M}_{T_n}(f)| \le \sum_{j=k}^{n} \frac{|G_j|}{|T_n \setminus T_{k-1}|} \bar{M}_{G_j}(f).
$$

Thus, it is enough to treat the case of $\bar{M}_{G_n}(f)$. In the case where all functions $f_S$ for $S \in \mathcal{N}_k$ are constant, writing $\bar{M}_{G_n}(f)$ as in (87), and using the convergence in distribution of $(S_{U_n})_{n \in \mathbb{N}}$ the uniform distribution over $\mathcal{N}_k$, we get that the sought convergence holds a.s. for $\bar{M}_{G_n}(f)$. Thus, without loss of generality, we assume that $\langle \pi, Q^S f_S \rangle = 0$ for all $S \in \mathcal{N}_k$.

Remark that it is enough to prove that $\sum_{n \ge k} \mathbb{E}[\bar{M}_{G_n}(f)^2] < \infty$, as then we can immediately conclude using Borel-Cantelli lemma with Markov's inequality. Thus, for $n \ge k$, using (92)
with $n' = |T_n|$ (such that $A_{n'} = G_n$) and $K = k$, we get:

$$
\begin{align*}
\mathbb{E}_Q[\bar{M}_{G_n}(f)^2] &\le C 2^{-(n-2k)} + C^{1/2} M 2^{-2n} \sum_{u,v \in G_n : d(u,v) > 2k} \alpha^{\bar{d}(u,v)-k} \\
&= C 2^{-(n-2k)} + C^{1/2} M \sum_{j=k}^{n} 2^{-(n-j)-1 \sum_{\{j>0\}} \alpha^{j-k}} \\
&\le C 2^{-(n-2k)} + C^{1/2} M 2^{-(n-k)} \sum_{j=k}^{n} 2^{j-k} \alpha^{j-k} \\
&\le C 2^{-(n-2k)} + C^{1/2} M 2^{-(n-k)} C' \max(n+1, (2\alpha)^{n-k}) \\
&\le C 2^{-(n-2k)} + C^{1/2} M C' \max((n+1)2^{-(n-k)}, \alpha^{n-k}),
\end{align*}
$$

where $C'$ is a constant whose value only depends on the value of $2\alpha$. Hence, there exist
finite constants $C_0 < \infty$ and $\beta \in (0, 1)$ such that (94) holds. In particular, we get that
$\sum_{n \ge k} \mathbb{E}[\bar{M}_{G_n}(f)^2] < \infty$. This concludes the proof of the theorem. $\square$

**Appendix B: Proof of the “backward” coupling Lemma 4.1**

We now prove Lemma 4.1.

*Proof of Lemma 4.1.* The proof relies on a “backward in time” bound from $u$ to $u \wedge v$, and then a “forward in time” bound from $u \wedge v$ to $v$. We divide the proof in two cases: first when $v$ is an ancestor of $u$, and then the general case.

For all $j \le k$, define the vertex $U_j = p^j(u)$ which is random for $j > h(u)$ (in which case, it depends on $\mathcal{U}$). Write $x_{U_k} = x$. For all $j \in \{1, \dots, k\}$, define the random set (which depends on $\mathcal{U}$):

$$
\Delta^{-}(u, k, j) = (\Delta^{*}(u, k) \setminus \{U_{k}\}) \cap (T^{\infty}(U_{k}) \setminus T^{\infty}(U_{j-1})),
$$

and in particular remark that we have $U_0 = u \notin \Delta^{-}(u, k, j)$ and $U_k = p^k(u) \notin \Delta^{-}(u, k, j)$.

**Case 1:** $v$ is an ancestor of $u$. We mimic the proof of (Cappé, Moulines and Rydén, 2005, Proposition 12.5.4). The proof of the first case relies on the observation that conditioned on $X_{p^k(u)}$ and $Y_{\Delta(u,k)}$, the backward ancestral process $X$ from $U_0 = u$ to $U_k = p^k(u)$ is a non-homogeneous Markov chain satisfying a uniform mixing condition. The fact that $(X_{U_j})_{0 \le j \le k}$ is a Markov chain comes from the Markov property of the HMT $(X, Y)$ (remind the discussion around (2) on page 3379) which gives for all $j \in \{1, \dots, k\}$:

$$
\begin{align}
\mathcal{L}(X_{U_j} | Y_{\Delta(u,k)}, X_{U_k}, X_{T^\infty(U_{j-1})}) &= \mathcal{L}(X_{U_j} | Y_{\Delta(u,k)}, X_{U_k}, X_{U_{j-1}}) \nonumber \\
&= \mathcal{L}(X_{U_j} | Y_{\Delta^{-}(u,k,j)}, X_{U_k}, X_{U_{j-1}}). \tag{95}
\end{align}
$$

For all integers $j \in \{1, \dots, k\}$, the backward transition kernel (which depends on $\mathcal{U}$) from $X_{U_{j-1}}$ to $X_{U_j}$ is defined as:

$$
\mathcal{B}_{x_{U_k}, j}[y_{\Delta(u,k)}](x_{U_{j-1}}; f) = \mathbb{E}_{\theta}[f(X_{U_j}) \mid Y_{\Delta(u,k)} = y_{\Delta(u,k)}, X_{U_k} = x_{U_k}, X_{U_{j-1}} = x_{U_{j-1}}],
$$

for any $x_{U_{j-1}} \in \mathcal{X}$ and any bounded Borel function $f$ on $\mathcal{X}$. By the Markov property (see (95)), note that $\mathcal{B}_{x_{U_k}, j}[y_{\Delta(u,k)}](x_{U_{j-1}}, f)$ only depends on $y_{\Delta^{-}(u,k,j)}$ instead of $y_{\Delta(u,k)}$, that is:

$$
\mathcal{B}_{x_{U_k}, j}[y_{\Delta(u,k)}](x_{U_{j-1}}; f)
$$
$$ 
\begin{aligned}
&= \mathbb{E}_{\theta} [f(X_{U_j}) \mid Y_{\Delta^{-}(u,k,j)} = y_{\Delta^{-}(u,k,j)}, X_{U_k} = x_{U_k}, X_{U_{j-1}} = x_{U_{j-1}}] \\
&= \frac{\int_{\mathcal{X}} f(x_{U_j}) p_{\theta}(y_{\Delta^{-}(u,k,j)}, x_{U_j} \mid X_{U_k} = x_{U_k}) q_{\theta}(x_{U_j}, x_{U_{j-1}}) \lambda(dx_{U_j})}{\int_{\mathcal{X}} p_{\theta}(y_{\Delta^{-}(u,k,j)}, x_{U_j} \mid X_{U_k} = x_{U_k}) q_{\theta}(x_{U_j}, x_{U_{j-1}}) \lambda(dx_{U_j})},
\end{aligned} 
\quad (96) $$

where:

$$ p_{\theta}(y_{\Delta^{-}(u,k,j)}, x_{U_j} \mid X_{U_k} = x_{U_k}) = \int_{\mathcal{X}^{\Delta^{-}(u,k,j)}} \prod_{w \in \Delta^{-}(u,k,j)} q_{\theta}(x_{\mathrm{p}(w)}, x_w) g_{\theta}(x_w, y_w) \prod_{w \in \Delta^{-}(u,k,j) \setminus \{U_j\}} \lambda(dx_w). $$

To simplify notations, we will keep the dependence on $y_{\Delta(u,k)}$ for all indices $j$. Note that the integral in the denominator in the right hand side of (96) is lower bounded by:

$$ \prod_{w \in \Delta^{-}(u,k,j)} \sigma^{-} \int_{\mathcal{X}} g_{\theta}(x_w, y_w) \lambda(dx_w), $$

and is thus positive $\mathbb{P}_{\theta}$-a.s. under Assumption 3.

Using Assumption 3, we get that those backward transition kernels satisfy the following Doeblin condition (remind Definition 2.6):

$$ \frac{\sigma^{-}}{\sigma^{+}} v_{x_{U_k}, j}[y_{\Delta(u,k)}](f) \leq B_{x_{U_k}, j}[y_{\Delta(u,k)}](x_{U_{j-1}}; f), $$

where for any bounded Borel function $f$ on $\mathcal{X}$, we have:

$$ 
\begin{aligned}
v_{x_{U_k}, j}[y_{\Delta(u,k)}](f) &= \mathbb{E}_{\theta}[f(X_{U_j}) \mid Y_{\Delta^{-}(u,k,j)} = y_{\Delta^{-}(u,k,j)}, X_{U_k} = x_{U_k}] \\
&= \frac{\int_{\mathcal{X}} f(x_{U_j}) p_{\theta}(y_{\Delta^{-}(u,k,j)}, x_{U_j} \mid X_{U_k} = x_{U_k}) \lambda(dx_{U_j})}{\int_{\mathcal{X}} p_{\theta}(y_{\Delta^{-}(u,k,j)}, x_{U_j} \mid X_{U_k} = x_{U_k}) \lambda(dx_{U_j})},
\end{aligned}
 $$

where note that the only difference with the definition of $B_{x_{U_k}, j}[y_{\Delta(u,k)}](x_{U_{j-1}}; f)$ is that the term $q_{\theta}(x_{U_j}, x_{U_{j-1}})$ has disappeared in both the numerator and the denominator of $v_{x_{U_k}, j}[y_{\Delta(u,k)}](f)$. Thus, Lemma 2.7 shows that the Dobrushin coefficient $\delta(B_{x_{U_k}, j})$ (defined in (4)) of the backward transition kernel $B_{x_{U_k}, j}$ is upper bounded by $\rho = 1 - \sigma^{-}/\sigma^{+}$.

Note that the Markov property in (95) (with $j = 1$) also gives us:

$$ 
\begin{aligned}
\mathcal{L}(X_{U_1} | Y_{\Delta(u,k)}, X_{U_k}, X_{U_0}) &= \mathcal{L}(X_{U_1} | Y_{\Delta^{-}(u,k,j)}, X_{U_k}, X_{U_0}) \\
&= \mathcal{L}(X_{U_1} | Y_{\Delta^{*}(u,k)}, X_{U_k}, X_{U_0}).
\end{aligned} 
\quad (97) $$

Finally, if we write:

$$ 
\begin{aligned}
\mathbb{P}_{\theta}(X_v \in \cdot \mid Y_{\Delta(u,k)} = y_{\Delta(u,k)}, X_{U_k} = x_{U_k}) \\
&= \int \mathbb{P}_{\theta}(X_v \in \cdot \mid Y_{\Delta^{-}(u,k,1)} = y_{\Delta^{-}(u,k,1)}, X_{U_k} = x_{U_k}, X_{U_1} = x_{U_1}) \\
&\quad \times \mathbb{P}_{\theta}(X_{U_1} \in dx_{U_1} \mid Y_{\Delta(u,k)} = y_{\Delta(u,k)}, X_{U_k} = x_{U_k}),
\end{aligned} 
\quad (98) $$

and we also write (using (97)):

$$ \mathbb{P}_{\theta}(X_v \in \cdot \mid Y_{\Delta^{*}(u,k)} = y_{\Delta^{*}(u,k)}, X_{U_k} = x_{U_k}) $$
$$ = \int \mathbb{P}_{\theta}(X_v \in \cdot | Y_{\Delta^{-}(u,k,1)} = y_{\Delta^{-}(u,k,1)}, X_{U_k} = x_{U_k}, X_{U_1} = x_{U_1}) \quad (99) \\ \times \mathbb{P}_{\theta}(X_{U_1} \in dx_{U_1} | Y_{\Delta^{*}(u,k)} = y_{\Delta^{*}(u,k)}, X_{U_k} = x_{U_k}), $$

then the two distributions (for $X_v$) on the left hand sides of those displayed equations can be considered as obtained through running $d(u, v) - 1$ iterations of the backward ancestral conditional Markov chain described above, using two different initial conditions. Therefore, as the Dobrushin coefficient is sub-multiplicative (remind Lemma 2.5), we get that those two probability distribution differ by at most $2 \rho^{d(u,v)-1}$ in total variation. This concludes the proof of the first case.

**Case 2: general case.** The proof of the second case relies on the observation that conditioned on $X_{p^k(u)}$ and $Y_{\Delta(u,k)}$, if we consider the process $X$ backward from $u$ to $u \wedge v$ (remind that $v \in \Delta^*(u, k)$) and then forward from $u \wedge v$ to $v$, we get a non-homogeneous Markov chain satisfying uniform mixing rate $\rho$. Note that as $v \in T^\infty(p^k(u), k) \setminus \{u\}$, we have that $u \wedge v \in \{U_1, \dots, U_k\}$. Using the first case, it only remains to check those observations for the forward segment, which were already proved in the proof of Lemma 3.2.

Hence, if we use the same decomposition as in (98) and (98), which corresponds to run $d(u, v)-1$ iterations of the backward-forward conditional chain described above ($d(u, u \wedge v)-1$ backward iterations and $d(u \wedge v, v)$ forward iterations), we get as in the first case that those two probability distribution differ by at most $2 \rho^{d(u,v)-1}$ in total variation. This concludes the proof of the lemma. $\square$

## Appendix C: Proof of (35) (used in the proof of Proposition 3.10)

Let $m \in \mathbb{N}^*$ be fixed through this section.

For ease of read, we restate some notation definitions used only in the proof of Proposition 3.10. For $u, v \in T^\infty$ with $h(u) \equiv h(v) \bmod m + 1$, we write $T(u, m) < T(v, m)$ if $u < v$. Moreover for $u, v \in T^\infty$, we write $u < T(v, m)$ if $h(u) < h(v)$ or $h(u) \le h(v) + m$ and for all $w \in T(v, m)$ with $h(w) = h(u)$ (note that such $w$ must exist), we have $u < w$. Informally $u$ is “above or on the left of $T(v, m)$”. For all $u \in T$, $k \in \mathbb{N}$, define the random subtrees which depend on $\mathcal{U}$:

$$ \Delta^*(T(u, m), k) = \bigcup \{T(v, m) : v \in \Delta^*(u, k(m+1)) \text{ such that } h(v) \equiv h(u) \bmod m+1\}, $$

and $\Delta(T(u, m), k) = \Delta^*(T(u, m), k) \cup T(u, m)$. When $h(u) \ge k(m+1)$, then those subtrees do not depend on $\mathcal{U}$, and we simply write $\Delta^*(T(u, m), k) = \Delta^*(T(u, m), k)$ and $\Delta(T(u, m), k) = \Delta(T(u, m), k)$ to indicate it. See Figure 5 on page 3396 for an illustration of the “past” subtree $\Delta^*(T(u, m), k)$ of the block subtree $T(u, m)$.

The goal of this section is to prove (35) for all $\theta \in \Theta$ and $x \in \mathcal{X}$, which we restate here for ease of read:

$$ \lim_{k \to \infty} \mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*} \left[ \log p_{\theta}(Y_{T_m} | Y_{\Delta^*(T_m, k)}, X_{p^{k(m+1)}}(\partial)) = x \right] = |T_m| \ell(\theta). $$
### C.1. Decomposition of the log-likelihood into subtree increments

Following (16), for all $u \in T$, $k \in \mathbb{N}$, $x \in \mathcal{X}$ and $\theta \in \Theta$, using the conditional probabilities formula, define:

$$ H_{T(u,m),k,x}(\theta) = \frac{p_{\theta}(Y_{\Delta(T(u,m),k)} | X_{p^k(u)} = x)}{p_{\theta}(Y_{\Delta^*(T(u,m),k)} | X_{p^k(u)} = x)} \quad (100) \\ = \int p_{\theta}(Y_{T(u,m)} | X_u = x_u) \mathbb{P}_{\theta}(X_u \in \mathrm{d}x_u | Y_{\Delta^*(T(u,m),k)}, X_{p^{(k-1)(m+1)}(u)} = x), $$

where:

$$ p_{\theta}(Y_{T(u,m)} | X_u = x_u) = \int_{\mathcal{X}^{|T(u,m)|}} g_{\theta}(x_u, Y_u) \prod_{w \in T(u,m) \setminus \{u\}} g_{\theta}(x_w, Y_w) q_{\theta}(x_{p(w)}, x_w) \lambda(\mathrm{d}x_w). $$

We then define the log-likelihood contribution of the subtree $T(u, m)$ with past over $k \in \mathbb{N}$ subtree generations (that is, $k(m+1)$ (node) generations) as:

$$ h_{T(u,m),k,x}(\theta) = \log H_{T(u,m),k,x}(\theta) $$

For all $n \in \mathbb{N}^*$, we decompose the tree $T_{n(m+1)-1}$ into subtrees of height $m$ (such as $T_m$), and we order those subtrees according to $<$. Hence, using (6), (7) and (100) and a telescopic sum argument, the log-likelihood of the observed variables $Y_{T_{n(m+1)-1}}$ can be rewritten as:

$$ \ell_{n(m+1)-1,x}(\theta) = \sum_{k=0}^{n-1} \sum_{u \in \mathcal{G}_{k(m+1)}} h_{T(u,m),k,x}(\theta). \quad (101) $$

### C.2. Construction of the log-likelihood increments with infinite past for subtrees

In this subsection, we construct the log-likelihood increments with infinite past for subtrees.

To construct the limit of the functions $h_{T(u,m),k,x}(\theta)$ we first prove the following lemma which states some uniform bound about the asymptotic behavior of those functions when $k \to \infty$.

**Lemma C.1** (Uniform bounds for $h_{T(u,m),k,x}(\theta)$). Assume that Assumptions 2–3 and 4-(ii) hold. For all vertices $u \in T$ and all integers $k, k' \in \mathbb{N}^*$, the following assertions hold true:

$$ \sup_{\theta \in \Theta} \sup_{x, x' \in \mathcal{X}} |h_{T(u,m),k,x}(\theta) - h_{T(u,m),k',x'}(\theta)| \le \frac{\rho^{(k \wedge k')(m+1)-1}}{(1-\rho)^{|T_m|}}, \quad (102) $$

$$ \sup_{\theta \in \Theta} \sup_{k \in \mathbb{N}^*} \sup_{x \in \mathcal{X}} |h_{T(u,m),k,x}(\theta)| \le (|T_m| \log b^+) \vee \left| \sum_{w \in T(u,m)} \log(\sigma^- b^-(Y_w)) \right|. \quad (103) $$

*Proof.* [The proof of the lemma is a straightforward adaptation of the proof of (Cappé, Moulines and Rydén, 2005, Lemma 12.3.2) using Lemma 3.2 for the coupling.] Let $k' \ge k \ge 1$, and write $v = p^{k(m+1)}(u)$, $v' = p^{k'(m+1)}(u)$. Then, write:

$$ H_{T(u,m),k,x}(\theta) = \int_{\mathcal{X}^2} \left[ \int_{\mathcal{X}^{T(u,m)}} \prod_{w \in T(u,m)} g_{\theta}(x_w, Y_w) q_{\theta}(x_{p(w)}, x_w) \lambda(\mathrm{d}x_w) \right] \quad (104) $$
$$ \times \mathbb{P}_{\theta}(X_{p(u)} \in \mathrm{d}x_{p(u)} | Y_{\Delta^*(T(u,m),k)}, X_v = x_v) \times \delta_x(\mathrm{d}x_v), $$

and using the Markov property at $X_v$, write:

$$ \begin{aligned} H_{T(u,m),k',x'}(\theta) = & \int_{\mathcal{X}^2} \left[ \int_{\mathcal{X}^{T(u,m)}} \prod_{w \in T(u,m)} g_{\theta}(x_w, Y_w) q_{\theta}(x_{p(w)}, x_w) \lambda(\mathrm{d}x_w) \right] \right. \\ & \left. \times \mathbb{P}_{\theta}(X_{p(u)} \in \mathrm{d}x_{p(u)} | Y_{\Delta^*(T(u,m),k)}, X_v = x_v) \right. \\ & \left. \times \mathbb{P}_{\theta}(X_v \in \mathrm{d}x_v | Y_{\Delta^*(T(u,m),k')} \setminus \Delta(T(u,m),k), X_{v'} = x'). \right] \end{aligned} \quad (105) $$

Applying Lemma 3.2, we get (note that the integrands in (104) and (105) are non-negative):

$$ \begin{aligned} & |H_{T(u,m),k,x}(\theta) - H_{T(u,m),k',x'}(\theta)| \\ & \le \rho^{k(m+1)-1} \sup_{x_{p(u)} \in \mathcal{X}} \int \prod_{w \in T(u,m)} g_{\theta}(x_w, Y_w) q_{\theta}(x_{p(w)}, x_w) \lambda(\mathrm{d}x_w) \\ & \le \rho^{k(m+1)-1} (\sigma^+)^{|T_m|} \prod_{w \in T(u,m)} \int g_{\theta}(x_w, Y_w) \lambda(\mathrm{d}x_w). \end{aligned} \quad (106) $$

The integral in (104) can be lower bounded giving us:

$$ H_{u,k,x}(\theta) \ge (\sigma^-)^{|T_m|} \prod_{w \in T(u,m)} \int g_{\theta}(x_w, Y_w) \lambda(\mathrm{d}x_w), \quad (107) $$

where the right hand side is positive by Assumption 3-(ii); and similarly for (105). Combining (106) with (107), and with the inequality $|\log x - \log y| \le |x - y|(x \wedge y)$, we get the first assertion of the lemma:

$$ |h_{T(u,m),k,x}(\theta) - h_{T(u,m),k',x'}(\theta)| \le \left(\frac{\sigma^+}{\sigma^-}\right)^{|T_m|} \rho^{k(m+1)-1} = \frac{\rho^{k(m+1)-1}}{(1-\rho)^{|T_m|}}. $$

Combining (100) and (107), we get:

$$ \prod_{w \in T(u,m)} \sigma^- b^-(Y_w) \le H_{T(u,m),k,x}(\theta) \le (b^+)^{|T_m|}, $$

which yields the second assertion of the lemma (remind that $b^-(Y_w) > 0$ for all $w \in T^\infty$ by Assumption 3-(ii)). □

We are now ready to construct the limit of the functions $h_{T(u,m),k,x}(\theta)$ and state some properties of this limit. Note that this result is stated for every $u \in T$, but we will only need it for $u = \partial$. Remind that we are in the stationary case, and that the HMT process $(X, Y)$ is defined on $T^\infty$.

**Proposition C.2 (Properties of the limit function $h_{T(u,m),\infty}(\theta)$).** Assume that Assumptions 1–4 hold. For every $u \in T$ and $\theta \in \Theta$, there exists $h_{T(u,m),\infty}(\theta) \in L^1(\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*})$ such that for all $x \in \mathcal{X}$, the sequence $(h_{T(u,m),k,x}(\theta))_{k \in \mathbb{N}}$ converges $\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*}$-a.s. and in $L^1(\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*})$ to $h_{T(u,m),\infty}(\theta)$.

Furthermore, this convergence is uniform over $\theta \in \Theta$ and $x \in \mathcal{X}$, that is, we have that $\lim_{k \to \infty} \sup_{\theta \in \Theta} \sup_{x \in \mathcal{X}} |h_{T(u,m),k,x}(\theta) - h_{T(u,m),\infty}(\theta)| = 0 \mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*}$-a.s. and in $L^1(\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*})$.
Note that the limit function $h_{T(u,m),\infty}(\theta)$ in the proposition can be interpreted as $\log p_{\theta}(Y_{T(u,m)} | Y_{\Lambda^*(T(u,m),\infty)})$, where $\Delta^*(T(u,m),\infty) = \{v \in T^\infty : v <_{\mathcal{U}} T(u,m)\}$ is a random subset of vertices. Note that $h_{T(u,m),\infty}(\theta)$ is a function of the random set of variables $(Y_v, v \in \Delta(T(u,m), \infty))$, where we define $\Delta(T(u,m), \infty) = \Delta^*(T(u,m), \infty) \cup T(u,m)$, and thus implicitly depend on $\mathcal{U}$ through $\Delta(T(u,m), \infty)$.

*Proof.* Fix some $u \in T$. Note that (102) shows that $(h_{T(u,m),k,x}(\theta))_{k \in \mathbb{N}}$ is Cauchy sequence uniformly in $\theta$ and $x$, and thus has $\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*}$-almost surely a limit when $k \to \infty$ which does not depend on $x$; we denote this limit by $h_{T(u,m),\infty}(\theta)$. Furthermore, we get from (103) that $(h_{T(u,m),k,x}(\theta))_{k \in \mathbb{N}}$ is uniformly bounded in $L^1(\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*})$, and thus $h_{T(u,m),\infty}(\theta)$ is in $L^1(\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*})$ and the convergence also holds in $L^1(\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*})$. Finally, as the bound in (102) is uniform in $\theta$ and $x$, we get that the convergence holds uniform over $\theta$ and $x$ both $\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*}$-almost surely and in $L^1(\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*})$. $\square$

### C.3. Properties of the contrast function

As the functions $h_{T(u,m),\infty}(\theta)$ are in $L^1(\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*})$ under the assumptions used in Proposition C.2, we can now define the contrast function $\ell^{(m)}$ (which is deterministic) for block subtree of height $m$ as:

$$ \ell^{(m)}(\theta) = \mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*} [h_{T_m, \infty}(\theta)], $$

where remind $\mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*}$ is the expectation corresponding to $\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*}$. We prove under the $L^2$ regularity assumption Assumption 5 the convergence of the normalized log-likelihood to this contrast function.

**Proposition C.3 (Ergodic convergence for the log-likelihood).** Assume that Assumptions 1–5 hold. Then, for all $x \in \mathcal{X}$, the normalized log-likelihood $|T_{n(m+1)-1}|^{-1}\ell_{n(m+1)-1,x}(\theta)$ converges $\mathbb{P}_{\theta^*}$-a.s. to the contrast function $\ell^{(m)}(\theta)$ as $n \to \infty$.

$$ \lim_{n \to \infty} \frac{|T_m|}{|T_{n(m+1)-1}|} \ell_{n(m+1)-1,x}(\theta) = \ell^{(m)}(\theta) \quad \mathbb{P}_{\theta^*}\text{-a.s.} \quad (108) $$

In particular, we get that $\ell^{(m)}(\theta) = |T_m|\ell(\theta)$.

*Proof.* Let $\theta \in \Theta$ be some parameter. Fix some $k \in \mathbb{N}^*$ and $x \in \mathcal{X}$. Remind (101). Applying (102) for each vertex $u \in G_{j(m+1)}$ with $j \in \{k, \dots, n-1\}$, we get:

$$ \frac{|T_m|}{|T_{n(m+1)-1}|} \left| \ell_{n(m+1)-1,x}(\theta) - \sum_{j=k}^{n-1} \sum_{u \in G_{j(m+1)}} h_{T(u,m),k,x}(\theta) \right| \\ \le \frac{\rho^{k(m+1)-1}}{(1-\rho)^{|T_m|}} + \frac{|T_m|}{|T_{n(m+1)-1}|} \sum_{j=0}^{k-1} \sum_{u \in G_{j(m+1)}} |h_{T(u,m),j,x}(\theta)|. \quad (109) $$

Note that by (103), we have that $|h_{T(u,m),j,x}(\theta)| < \infty \mathbb{P}_{\theta^*}$-a.s. for all $j \in \mathbb{N}^*$ and $u \in G_{j(m+1)}$. For $u = \partial$, we have $h_{T_m,0,x}(\theta) = \log p_{\theta}(Y_{T_m} | X_{\partial} = x)$ which is finite $\mathbb{P}_{\theta^*}$-a.s. by Assumption 3-(iii).

The definition of the shape from Section 2.4 can straightforwardly be adapted to the (deterministic) subtrees $\Delta(T(u,m), k)$ for vertices $u \in G_{j(m+1)}$ with $j \ge k$, where $u$ is seen
as a distinguished vertex of $\Delta(T(u, m), k)$. Following (8) (on page 3383) in the initial vertex-by-vertex decomposition setting, for a vertex $u \in G_{j(m+1)}$ with $j \ge k$, let $v_u \in G_{k(m+1)}$ be the unique vertex $G_{k(m+1)}$ such that $\Delta(T(u, m), k)$ and $\Delta(T(v_u, m), k)$ have the same shape. Then, we have:

$$ \begin{aligned} & h_{T(u,m),k,x}(\theta; Y_{\Delta(T(u,m),k)}) = y_{\Delta(T(u,m),k)}) \\ & = h_{T(v_u,m),k,x}(\theta; Y_{\Delta(T(v_u,m),k)}) = y_{\Delta(T(v_u,m),k))}. \end{aligned} \quad (110) $$

Moreover, using (103) together with Assumption 5, we get for every $u \in G_{j(m+1)}$ with $j \ge k$ that the random variable $h_{T(u,m),k,x}(\theta; Y_{\Delta(T(u,m),k)})$ is in $L^2(\mathbb{P}_{\theta^*})$. Hence, applying a straightforward modification of Lemma 2.11 for subtree blocks $T(u, m)$ to the collection of neighborhood-shape-dependent functions $(h_{T(v,m),k,x}(\theta; Y_{\Delta(T(v,m))}) = \cdot)_{v \in G_{k(m+1)}}$ (remind that indexing functions with $G_{k(m+1)}$ or with the set of possible shapes is equivalent by (9)), and using (110) and (14) (in Remark 3.1), we get:

$$ \frac{|T_m|}{|T_{n(m+1)-1}|} \sum_{j=k}^{n-1} \sum_{u \in G_{j(m+1)}} h_{T(u,m),k,x}(\theta) \underset{n \to \infty}{\longrightarrow} \mathbb{E}_{u} \otimes \mathbb{E}_{\theta^*} [h_{T_m,k,x}(\theta)] \quad \mathbb{P}_{\theta^*\text{-a.s.}} \quad (111) $$

Using (102) with Proposition C.2, we get:

$$ |\mathbb{E}_{u} \otimes \mathbb{E}_{\theta^*} [h_{T_m,k,x}(\theta)] - \mathbb{E}_{u} \otimes \mathbb{E}_{\theta^*} [h_{T_m,\infty}(\theta)]| \le \frac{\rho^{k(m+1)-1}}{(1-\rho)^{|T_m|}}. $$

Thus, combining this bound with (109) and (111), we get $\mathbb{P}_{\theta^*\text{-a.s.}}$ that:

$$ \limsup_{n \to \infty} \left| \frac{|T_m|}{|T_{n(m+1)-1}|} \ell_{n(m+1)-1,x}(\theta) - \mathbb{E}_{u} \otimes \mathbb{E}_{\theta^*} [h_{T_m,\infty}(\theta)] \right| \le 2 \frac{\rho^{k(m+1)-1}}{(1-\rho)^{|T_m|}}. $$

As the left hand side does not depend on $k$, letting $k \to \infty$, we get that (108) in the lemma holds. Lastly, as the limit must be the same as in Proposition 3.5, we get that $\ell^{(m)}(\theta) = |T_m|\ell(\theta)$. This concludes the proof. $\square$

We are now ready to close this section by proving that (35) holds.

**Proposition C.4.** Assume that Assumptions 1–5 hold. Then, (35) holds for all $\theta \in \Theta$ and $x \in \mathcal{X}$.

*Proof.* Applying Proposition C.2, we get that the left hand side of (35) is equal to $\ell^{(m)}(\theta) = \mathbb{E}_{u} \otimes \mathbb{E}_{\theta^*} [h_{T(\partial,m),\infty}(\theta)]$, which is equal to $|T_m|\ell(\theta)$ by Proposition C.3. $\square$

## Appendix D: Details of the proof of Proposition 4.5

Remind that the proof of Proposition 4.4 can be straightforwardly adapted to Proposition 4.5 except for Lemma 4.8. In Section 4.2.2, for brevity, we have only presented the adaptation of Lemma 4.8 to the terms $\Gamma_{u,k,x}(\theta)$. In this appendix, we present all the details of the adaptation of the rest of the proof of Proposition 4.4 to the terms $\Gamma_{u,k,x}(\theta)$.

The following lemma gives an exponential bound on the $L^2(\mathbb{P}_{\theta^*})$ norm uniformly in $x \in \mathcal{X}$ for the the average of the quantities $\Gamma_{u,h(u),x}(\theta^*)$ over $u \in T_n^*$.
**Lemma D.1.** Under the assumptions of Proposition 4.5, for all $x \in \mathcal{X}$ and $\theta \in \Theta_0$, there exist finite constants $C < \infty$ and $\alpha \in (0, 1)$ such that for all $n \in \mathbb{N}^*$ we have:

$$ \mathbb{E}_{\theta^*} \left[ \sup_{x \in \mathcal{X}} \left| \frac{1}{|T_n|} \sum_{u \in T_n^*} \Gamma_{u,h(u),x}(\theta) - \mathbb{E}u \otimes \mathbb{E}_{\theta^*} [\Gamma_{\partial,\infty}(\theta)] \right|^2 \right]^{1/2} \le C \alpha^n. \quad (112) $$

*Proof.* Let $x' \in \mathcal{X}$ and $\theta \in \Theta_0$. Using Minkowski's inequality and Jensen's inequality, for all $n, k \in \mathbb{N}^*$, we get:

$$ \begin{aligned} & \mathbb{E}_{\theta^*} \left[ \sup_{x \in \mathcal{X}} \left| \frac{1}{|T_n|} \sum_{u \in T_n^*} \Gamma_{u,h(u),x}(\theta) - \mathbb{E}u \otimes \mathbb{E}_{\theta^*} [\Gamma_{\partial,\infty}(\theta)] \right|^2 \right]^{1/2} \\ & \le \mathbb{E}_{\theta^*} \left[ \sup_{x,x' \in \mathcal{X}} \left| \frac{1}{|T_n|} \sum_{u \in T_{k-1}^*} \Gamma_{u,h(u),x}(\theta) \right|^2 \right]^{1/2} \\ & \quad + \mathbb{E}_{\theta^*} \left[ \sup_{x,x' \in \mathcal{X}} \left| \frac{1}{|T_n|} \sum_{u \in T_n \setminus T_{k-1}} \Gamma_{u,h(u),x}(\theta) - \Gamma_{u,k,x'}(\theta) \right|^2 \right]^{1/2} \\ & \quad + \mathbb{E}_{\theta^*} \left[ \left| \frac{1}{|T_n|} \sum_{u \in T_n \setminus T_{k-1}} \Gamma_{u,k,x'}(\theta) - \mathbb{E}u \otimes \mathbb{E}_{\theta^*} [\Gamma_{\partial,k,x'}(\theta)] \right|^2 \right]^{1/2} \\ & \quad + \mathbb{E}u \otimes \mathbb{E}_{\theta^*} [ |\Gamma_{\partial,k,x'}(\theta) - \Gamma_{\partial,\infty}(\theta)|^2 ]^{1/2}. \end{aligned} \quad (113) $$

Using Lemma 4.17 together with (49) on page 3403 (which, remind, are both immediate consequences of Lemma 4.2), there exists a finite constant $C < \infty$ and $\beta \in (0, 1)$ such that the first term in the right hand side of (113) is upper bounded by $C2^{-(n-k)}$ (note that $|T_{k-1}|/|T_n| \le 2^{-(n-k)}$), and the second and fourth terms in the right hand side of (113) are both upper bounded by $C\beta^{k/2}$.

We now give an upper bound for the second term in the right hand side of (113). For a vertex $u$ in $T \setminus T_{k-1}$, let $v_u \in G_k$ be the unique vertex that satisfies the shape equality constraint (8) (on page 3383), then we have:

$$ \Gamma_{u,k,x'}(\theta; Y_{\Delta(u,k)} = y_{\Delta(u,k)}) = \Gamma_{v_u,k,x'}(\theta; Y_{\Delta(v_u)} = y_{\Delta(u,k)}). \quad (114) $$

Moreover, using the definition of $\Gamma_{u,k,x}(\theta)$ in (63) together with the assumption on $\phi_\theta$ in Proposition 4.5, we get that the random variable $\Gamma_{u,k,x'}(\theta; Y_{\Delta(u,k)} = y_{\Delta(u,k)})$ is in $L^2(\mathbb{P}_{\theta^*})$ for every $u \in T \setminus T_{k-1}$. Thus, we can apply Lemma 2.11 (see in particular (11)) to the collection of neighborhood-shape-dependent functions $(\Gamma_{v_u,k,x'}(\theta; Y_{\Delta(v)} = \cdot))_{v \in G_k}$ (remind that indexing functions with $G_k$ or with $N_k$ is equivalent by (9)). Using (11) in Lemma 2.11 together with (28) and (14) in Remark 3.1, we get that there exist $\gamma \in (0, 1)$ and a finite constant $C' < \infty$ (note that they both do not depend on $k$ and $n$) such that for all $n, k \in \mathbb{N}^*$ with $n \ge k$, the second term in the right hand side of (113) is upper bounded by $C'\gamma^{n-k}$.
Hence, taking $k = \lceil n/2 \rceil$, we get that the left hand side of (113) is upper bounded by $2C\beta^{n/4} + C'\alpha^{n/2} + C2^{-n/2+1}$, and thus decays at exponential rate as desired. This concludes the proof. $\square$

Lemma D.1 implies as a corollary the convergence $\mathbb{P}_{\theta^*}$-a.s. and in $L^2(\mathbb{P}_{\theta^*})$ uniformly in $x \in \mathcal{X}$ for the sum of the quantities $\Gamma_{u,h(u),x}(\theta^*)$ over $u \in T_n^*$.

**Corollary D.2.** *Under the assumptions of Proposition 4.5, for all $x \in \mathcal{X}$ and $\theta \in \Theta_0$, we have:*

$$ \lim_{n \to \infty} \sup_{x \in \mathcal{X}} \left| \frac{1}{|T_n|} \sum_{u \in T_n^*} \Gamma_{u,h(u),x}(\theta) - \mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*} [\Gamma_{\partial,\infty}(\theta)] \right| = 0, \quad \mathbb{P}_{\theta^*}\text{-a.s. and in } L^2(\mathbb{P}_{\theta^*}). $$

*Proof.* The convergence in $L^2(\mathbb{P}_{\theta^*})$ follows immediately from Lemma D.1. Moreover, using again Lemma D.1, we have:

$$ \sum_{n \in \mathbb{N}^*} \mathbb{E}_{\theta^*} \left[ \sup_{x \in \mathcal{X}} \left| \frac{1}{|T_n|} \sum_{u \in T_n^*} \Gamma_{u,\infty}(\theta) - \mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*} [\Gamma_{\partial,\infty}(\theta)] \right|^2 \right] < \infty. $$

Hence, Borel-Cantelli lemma and Markov's inequality imply that the convergence in the lemma also holds $\mathbb{P}_{\theta^*}$-a.s. $\square$

The following lemma gives some continuity properties of the function $\theta \mapsto \Gamma_{\partial,k,x}(\theta)$.

**Lemma D.3.** *Under the assumptions of Proposition 4.5, for all $x \in \mathcal{X}$ and $k \in \mathbb{N}$, the random function $\theta \mapsto \Gamma_{\partial,k,x}(\theta)$ is $\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*}$-a.s. continuous on $\Theta_0$. Moreover, for all $\theta \in \Theta_0$, we have:*

$$ \lim_{\delta \to 0} \mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*} \left[ \sup_{\theta' \in \Theta_0 : \|\theta' - \theta\| \le \delta} |\Gamma_{\partial,k,x}(\theta') - \Gamma_{\partial,k,x}(\theta)|^2 \right] = 0. $$

*Proof.* We mimic the proof of (Douc, Moulines and Rydén, 2004, Lemma 14).

For all $v \in T^\infty$, define the random variable:

$$ \|\phi^v\|_{\infty} = \sup_{\theta' \in \Theta_0} \sup_{x,x' \in \mathcal{X}} |\phi_{\theta'}(x', x, Y_v)|. $$

Remind that under the assumptions of Proposition 4.5, the HMT process $(X, Y)$ is stationary and the random variable $\|\phi^\partial\|_{\infty}$ is in $L^4(\mathbb{P}_{\theta^*})$. Thus, for all $v \in T^\infty$, the random variable $\|\phi^v\|_{\infty}$ is in $L^4(\mathbb{P}_{\theta^*})$. Remind from (12) on page 3386 that $\Delta(\partial, k)$ is a random subtree of the deterministic subtree $T^\infty(p^k(u), k)$. Then, note that we have:

$$ \sup_{\theta \in \Theta_0} |\Gamma_{\partial,k,x}(\theta)| \le 4 \left( \sum_{v \in T^\infty(p^k(\partial), k)} \|\phi^v\|_{\infty} \right)^2, $$

where the upper bound is a random variable in $L^2(\mathbb{P}_{\theta^*})$ (and thus in $L^2(\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*})$) which depends on $Y_{T^\infty(p^k(u),k)}$ but not on $\mathcal{U}$. Hence, to prove the lemma, it suffices to prove that for all $v_1, v_2 \in T^\infty(p^k(u), k) \setminus \{p^k(\partial)\}$ and $\epsilon \in \{0, 1\}$, we have $\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*}$-a.s.:
$$
\lim_{\delta \to 0} \sup_{\theta' \in \Theta_0 : \|\theta' - \theta\| \le \delta} \left| \mathbb{E}_{\theta'} [\phi_{\theta', v_1, v_2}^{(2, \epsilon)} | Y_{\Delta(\partial, k)}, X_{p^k(\partial)} = x] - \mathbb{E}_{\theta} [\phi_{\theta, v_1, v_2}^{(2, \epsilon)} | Y_{\Delta(\partial, k)}, X_{p^k(\partial)} = x] \right| = 0,
$$

where:

$$
\phi_{\theta', v_1, v_2}^{(2, \epsilon)} := \phi_{\theta'}(X_{p(v_1)}, X_{v_1}, Y_{v_1}) \phi_{\theta'}(X_{p(v_2)}, X_{v_2}, Y_{v_2})^{\epsilon}.
$$

Denote $x_{p^k(\partial)} = x$, and write:

$$
\mathbb{E}_{\theta} [\phi_{\theta', v_1, v_2}^{(2, \epsilon)} | Y_{\Delta(\partial, k)}, X_{p^k(\partial)} = x] \\
= \frac{\int_{\mathcal{X}^{\Delta(\partial,k)} \setminus \{p^k(\partial)\}} \phi_{\theta'}^{(2, \epsilon)} (x_{p(v_1)}, x_{v_1}, Y_{v_1}, x_{p(v_2)}, x_{v_2}, Y_{v_2}) \Psi(dx_{\Delta(\partial,k)} \setminus \{p^k(\partial)\})}{\int_{\mathcal{X}^{\Delta(\partial,k)} \setminus \{p^k(\partial)\}} 1 \Psi(dx_{\Delta(\partial,k)} \setminus \{p^k(\partial)\})} . \quad (115)
$$

where:

$$
\Psi(dx_{\Delta(\partial,k)} \setminus \{p^k(\partial)\}) := \prod_{w \in \Delta(\partial,k) \setminus \{p^k(\partial)\}} q_{\theta}(x_{p(w)}, x_w) g_{\theta}(x_w, Y_w) \lambda(dx_w).
$$

Using Assumptions 2–4 (which are part of the assumptions in Proposition 4.5), we know that the integrand in the numerator of the right hand side of (115) is continuous w.r.t. $\theta$ and is upper bounded by the random variable $\|\phi^{v_1}\|_{\infty}(\|\phi^{v_2}\|_{\infty})^{\epsilon}(\sigma^+ b^+)^{|T^{\infty}(p^k(u),k)|-1}$ (remind that $\sigma^+ \ge 1$ and $b^+ \ge 1$). And similarly, the denominator is continuous w.r.t. $\theta$, and, using Assumption 3-(ii), is lower bounded by the random variable:

$$
\prod_{w \in \Delta(\partial, k) \setminus \{p^k(\partial)\}} \sigma^{-} \inf_{\theta' \in \Theta} \int g_{\theta'}(x_w, Y_w) \lambda(dx_w) > 0.
$$

Hence, using dominated convergence, we conclude that $P_u \otimes P_{\theta^*}$-a.s. the left hand side of (115) is continuous w.r.t. $\theta$. This concludes the proof. $\square$

As a corollary of Lemma D.3, we get that the function $\theta \mapsto \Gamma_{\partial,\infty}(\theta)$ is continuous in $L^2(\mathbb{P}_{\theta^*})$.

**Corollary D.4.** Under the assumptions of Proposition 4.5, for all $\theta \in \Theta_0$, we have:

$$
\lim_{\delta \to 0} \mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*} \left[ \sup_{\theta' \in \Theta_0 : \|\theta' - \theta\| \le \delta} |\Gamma_{\partial, \infty}(\theta') - \Gamma_{\partial, \infty}(\theta)|^2 \right] = 0.
$$

In particular, the function $\theta \mapsto \mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*}[\Gamma_{\partial, \infty}(\theta)]$ is continuous on $\Theta_0$.

*Proof.* Using Minkowski's inequality and Lemma 4.17, there exist a finite constant $C < \infty$ and $\beta \in (0, 1)$ such that for all $x \in \mathcal{X}$ and $k \in \mathbb{N}^*$, we have:

$$
\mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*} \left[ \sup_{\theta' \in \Theta_0 : \|\theta' - \theta\| \le \delta} |\Gamma_{\partial, \infty}(\theta') - \Gamma_{\partial, \infty}(\theta)|^2 \right]^{1/2} \\
\le 2C\beta^{k/2} + \mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*} \left[ \sup_{\theta' \in \Theta_0 : \|\theta' - \theta\| \le \delta} |\Gamma_{\partial, k, x}(\theta') - \Gamma_{\partial, k, x}(\theta)|^2 \right]^{1/2} . \quad (116)
$$
Using Lemma D.3, we get:

$$ \limsup_{\delta \to 0} \mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*} \left[ \sup_{\theta' \in \Theta_0 : \|\theta' - \theta\| \le \delta} \left| \Gamma_{\partial, \infty}(\theta') - \Gamma_{\partial, \infty}(\theta) \right|^2 \right]^{1/2} \le 2C\beta^{k/2}, $$

and taking $k \to \infty$, the upper bound vanishes. This concludes the proof. □

We now prove a locally uniform law of large numbers for the quantities $\Gamma_{u,k,x}(\theta)$.

**Lemma D.5.** Under the assumptions of Proposition 4.5, for all $x \in \mathcal{X}$, we have:

$$ \lim_{\delta \to 0} \lim_{n \to \infty} \sup_{\theta' \in \Theta_0 : \|\theta' - \theta\| \le \delta} \left| \frac{1}{|T_n|} \sum_{u \in T_n^*} \Gamma_{u,h(u),x}(\theta') - \mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*} [\Gamma_{\partial, \infty}(\theta)] \right| = 0, \quad \mathbb{P}_{\theta^*\text{-a.s.}} $$

*Proof.* First, write:

$$ \sup_{\theta' \in \Theta_0 : \|\theta' - \theta\| \le \delta} \left| \frac{1}{|T_n|} \sum_{u \in T_n^*} \Gamma_{u,h(u),x}(\theta') - \mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*} [\Gamma_{\partial, \infty}(\theta)] \right| \\ \le \frac{1}{|T_n|} \sum_{u \in T_n^*} \sup_{\theta' \in \Theta_0 : \|\theta' - \theta\| \le \delta} \left| \Gamma_{u,h(u),x}(\theta') - \Gamma_{u,h(u),x}(\theta) \right| \\ + \left| \frac{1}{|T_n|} \sum_{u \in T_n^*} \Gamma_{u,h(u),x}(\theta) - \mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*} [\Gamma_{\partial, \infty}(\theta)] \right|. \quad (117) $$

Then, we use the exact same argument as in the proofs of Lemma D.1 and Corollary D.2 where for all $u \in T^*$, the random variable $\Gamma_{u,k,x}(\theta)$ is replaced by the random variable:

$$ \sup_{\theta' \in \Theta_0 : \|\theta' - \theta\| \le \delta} \left| \Gamma_{u,h(u),x}(\theta') - \Gamma_{u,h(u),x}(\theta) \right|, $$

which are in $L^2(\mathbb{P}_{\theta^*})$ using the assumptions of Proposition 4.5. This gives us that the first term in the upper bound of (117) converges $\mathbb{P}_{\theta^*\text{-a.s.}}$ as $n \to \infty$ to:

$$ \mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*} \left[ \sup_{\theta' : \|\theta' - \theta\| \le \delta} \left| \Gamma_{\partial, \infty}(\theta') - \Gamma_{\partial, \infty}(\theta) \right| \right], $$

which, by Corollary D.4, vanishes when $\delta \to 0$. Corollary D.2 implies that the second term in the upper bound of (117) vanishes $\mathbb{P}_{\theta^*\text{-a.s.}}$ when $n \to \infty$. This concludes the proof. □

Combining the previous lemmas in this appendix and Lemma 4.17, we are now ready to prove Proposition 4.5.

**Proof of Proposition 4.5.** Applying Lemma 4.17, for all $u \in T$, we have that the sequence $(\Gamma_{u,k,x}(\theta))_{k \in \mathbb{N}^*}$ is a Cauchy sequence uniformly w.r.t. $\theta \in \Theta_0$ in $L^2(\mathbb{P}_{\mathcal{U}} \otimes \mathbb{P}_{\theta^*})$ that converges to some limit $\Gamma_{u,\infty}(\theta)$ (that does not depend on $x$). By Corollary D.2, we have that $\mathbb{P}_{\theta^*\text{-a.s.}}$ the convergence for the the average of the quantities $\Gamma_{u,h(u),x}(\theta^*)$ over $u \in T_n^*$ holds uniformly in $x \in \mathcal{X}$, that is, (66) in Proposition 4.5 holds. By Corollary D.4, we have that the function $\theta \mapsto \mathbb{E}_{\mathcal{U}} \otimes \mathbb{E}_{\theta^*} [\Gamma_{\partial, \infty}(\theta)]$ is continuous on $\Theta_0$. Finally, the last part of the proposition is given by Lemma D.5. □
Acknowledgments

The author would like to thank an anonymous referee for their constructive comments that
helped improve the quality of this paper.

References

ATHREYA, K. B. (1969). Limit Theorems for Multitype Continuous Time Markov Branching Processes: I. The Case of an Eigenvector Linear Functional. *Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete* **12** 320–332. https://doi.org/10.1007/BF00538753 MR0254927

BAUM, L. E. and PETRIE, T. (1966). Statistical Inference for Probabilistic Functions of Finite State Markov Chains. *The Annals of Mathematical Statistics* **37** 1554–1563. https://doi.org/10.1214/aoms/1177699147 MR0202264

BAUM, L. E., PETRIE, T., SOULES, G. and WEISS, N. (1970). A Maximization Technique Occurring in the Statistical Analysis of Probabilistic Functions of Markov Chains. *The Annals of Mathematical Statistics* **41** 164–171. https://doi.org/10.1214/aoms/1177697196 MR0287613

BERCU, B., DE SAPORTA, B. and GÉGOUT-PETIT, A. (2009). Asymptotic Analysis for Bifurcating AutoRegressive Processes via a Martingale Approach. *Electronic Journal of Probability* **14**. https://doi.org/10.1214/EJP.v14-717 MR2563249

BERTSEKAS, D. P. and SHREVE, S. E. (1996). *Stochastic Optimal Control: The Discrete Time Case. Optimization and Neural Computation Series*. Athena Scientific, Belmont, Mass. MR4496006

BICKEL, P. J., RITOV, Y. and RYDÉN, T. (1998). Asymptotic Normality of the Maximum-Likelihood Estimator for General Hidden Markov Models. *The Annals of Statistics* **26**. https://doi.org/10.1214/aos/1024691255 MR1647705

BIESINGER, J., WANG, Y. and XIE, X. (2013). Discovering and Mapping Chromatin States Using a Tree Hidden Markov Model. *BMC Bioinformatics* **14** S4. https://doi.org/10.1186/1471-2105-14-S5-S4

BILLINGSLEY, P. (1999). *Convergence of Probability Measures*, 2nd ed. Wiley Series in Probability and Statistics. Probability and Statistics Section. Wiley, New York. MR1700749

BITSEKI PENDA, S. V. and DELMAS, J.-F. (2022a). Central Limit Theorem for Bifurcating Markov Chains under Pointwise Ergodic Conditions. *The Annals of Applied Probability* **32**. MR4497859

BITSEKI PENDA, S. V. and DELMAS, J.-F. (2022b). Central Limit Theorem for Kernel Estimator of Invariant Density in Bifurcating Markov Chains Models. *Journal of Theoretical Probability*. MR4621077

BITSEKI PENDA, S. V., DJELLOUT, H. and GUILLIN, A. (2014). Deviation Inequalities, Moderate Deviations and Some Limit Theorems for Bifurcating Markov Chains with Application. *The Annals of Applied Probability* **24**. https://doi.org/10.1214/13-AAP921 MR3161647

BOGACHEV, V. I. (2007). *Measure Theory. Vol. 2*. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34514-5 MR2267655
BOUGUILA, N., FAN, W. and AMAYRI, M., eds. (2022). *Hidden Markov Models and Applications*. Unsupervised and Semi-Supervised Learning. Springer International Publishing, Cham. https://doi.org/10.1007/978-3-030-99142-5 MR4442254

CAPPÉ, O., MOULINES, É. and RYDÉN, T. (2005). *Inference in Hidden Markov Models*. Springer Series in Statistics. Springer New York, New York, NY. https://doi.org/10.1007/0-387-28982-8 MR2159833

CHOI, H. and BARANIUK, R. G. (2001). Multiscale Image Segmentation Using Wavelet-Domain Hidden Markov Models. *IEEE Transactions on Image Processing* **10** 1309–1321. https://doi.org/10.1109/83.941855 MR1852250

CROUSE, M. S., NOWAK, R. D. and BARANIUK, R. G. (1998). Wavelet-Based Statistical Signal Processing Using Hidden Markov Models. *IEEE Transactions on Signal Processing* **46** 886–902. https://doi.org/10.1109/78.668544 MR1665651

DELMAS, J.-F. and MARSALLE, L. (2010). Detection of Cellular Aging in a Galton-Watson Process. *Stochastic Processes and their Applications* **120** 2495–2519. https://doi.org/10.1016/j.spa.2010.07.002 MR2728175

DEMPSTER, A. P., LAIRD, N. M. and RUBIN, D. B. (1977). Maximum Likelihood from Incomplete Data Via the EM Algorithm. *Journal of the Royal Statistical Society Series B: Statistical Methodology* **39** 1–22. https://doi.org/10.1111/j.2517-6161.1977.tb01600.x MR0501537

DOUC, R. and MATIAS, C. (2001). Asymptotics of the Maximum Likelihood Estimator for General Hidden Markov Models. *Bernoulli* **7** 381. https://doi.org/10.2307/3318493 MR1836737

DOUC, R., MOULINES, É. and RYDÉN, T. (2004). Asymptotic Properties of the Maximum Likelihood Estimator in Autoregressive Models with Markov Regime. *The Annals of Statistics* **32**. https://doi.org/10.1214/009053604000000021 MR2102510

DOUC, R., ROUEFF, F. and SIM, T. (2016). The Maximizing Set of the Asymptotic Normalized Log-Likelihood for Partially Observed Markov Chains. *The Annals of Applied Probability* **26**. https://doi.org/10.1214/15-AAP1149 MR3543899

DOUC, R., MOULINES, E., OLSSON, J. and VAN HANDEL, R. (2011). Consistency of the Maximum Likelihood Estimator for General Hidden Markov Models. *The Annals of Statistics* **39**. https://doi.org/10.1214/10-AOS834 MR2797854

DOUC, R., MOULINES, É., PRIOURET, P. and SOULIER, P. (2018). *Markov Chains*. Springer Series in Operations Research and Financial Engineering. Springer International Publishing, Cham. MR3889011

DUARTE, M. F., WAKIN, M. B. and BARANIUK, R. G. (2008). Wavelet-Domain Compres- sive Signal Reconstruction Using a Hidden Markov Tree Model. In *2008 IEEE International Conference on Acoustics, Speech and Signal Processing* 5137–5140. IEEE, Las Vegas, NV, USA. https://doi.org/10.1109/ICASSP.2008.4518815

DUFLO, M. (2011). *Random Iterative Models*. Springer, Berlin. MR1485774

DURAND, J.-B., GONÇALVES, P. and GUÉDON, Y. (2004). Computational Methods for Hidden Markov Tree Models—An Application to Wavelet Trees. *IEEE Transactions on Signal Processing* **52** 2551–2560. https://doi.org/10.1109/TSP.2004.832006 MR2091804

DURAND, J.-B., GUÉDON, Y., CARAGLIO, Y. and COSTES, E. (2005). Analysis of the Plant Architecture via Tree-structured Statistical Models: The Hidden Markov Tree Models. *New*
Phytologist **166** 813–825. https://doi.org/10.1111/j.1469-8137.2005.01405.x

GENON-CATALOT, V. and LAREDO, C. (2006). Leroux's Method for General Hidden Markov Models. *Stochastic Processes and their Applications* **116** 222–243. https://doi.org/10.1016/j.spa.2005.10.005 MR2197975

GRAVE, É., OBOZINSKI, G. and BACH, F. (2013). Hidden Markov Tree Models for Semantic Class Induction. In *CoNLL Proceedings of the Seventeenth Conference on Computational Natural Language Learning* 94–103. Association for Computational Linguistics, Sofia, Bulgaria.

GUYON, J. (2007). Limit Theorems for Bifurcating Markov Chains. Application to the Detection of Cellular Aging. *The Annals of Applied Probability* **17** 1538–1569. https://doi.org/10.1214/105051607000000195 MR2358633

HANZOULI-BEN SALAH, H., LAPUYADE-LAHORGUE, J., BERT, J., BENOIT, D., LAMBIN, P., VAN BAARDWIJK, A., MONFRINI, E., PIECZYNSKI, W., VISVIKIS, D. and HATT, M. (2017). A Framework Based on Hidden Markov Trees for Multimodal Image Co-segmentation. *Medical Physics* **44** 5835–5848. https://doi.org/10.1002/mp.12531

HU, K., YANG, W. and GAO, X. (2017). Microcalcification Diagnosis in Digital Mammography Using Extreme Learning Machine Based on Hidden Markov Tree Model of Dual-Tree Complex Wavelet Transform. *Expert Systems with Applications* **86** 135–144. https://doi.org/10.1016/j.eswa.2017.05.062

JENSEN, J. L. and PETERSEN, N. V. (1999). Asymptotic Normality of the Maximum Likelihood Estimator in State Space Models. *The Annals of Statistics* **27**. https://doi.org/10.1214/aos/1018031205 MR1714719

KASAHARA, H. and SHIMOTSU, K. (2019). Asymptotic Properties of the Maximum Likelihood Estimator in Regime Switching Econometric Models. *Journal of Econometrics* **208** 442–467. https://doi.org/10.1016/j.jeconom.2018.09.019 MR3913246

KONDO, S., DUH, K. and MATSUMOTO, Y. (2013). Hidden Markov Tree Model for Word Alignment. In *WMT Proceedings of the Eighth Workshop on Statistical Machine Translation* 503–511. Association for Computational Linguistics, Sofia, Bulgaria.

KOSKI, T. (2001). *Hidden Markov Models for Bioinformatics*. *Computational Biology* **2**. Kluwer academic publ, Dordrecht [etc.]. MR1888250

LE GLAND, F. and MEVEL, L. (2000). Exponential Forgetting and Geometric Ergodicity in Hidden Markov Models. *Mathematics of Control, Signals, and Systems* **13** 63–93. https://doi.org/10.1007/PL00009861 MR1742140

LEROUX, B. G. (1992). Maximum-Likelihood Estimation for Hidden Markov Models. *Stochastic Processes and their Applications* **40** 127–143. https://doi.org/10.1016/0304-4149(92)90141-C MR1145463

MAKHIJANI, M. K., BALU, N., YAMADA, K., YUAN, C. and NAYAK, K. S. (2012). Accelerated 3D MERGE Carotid Imaging Using Compressed Sensing with a Hidden Markov Tree Model. *Journal of Magnetic Resonance Imaging* **36** 1194–1202. https://doi.org/10.1002/jmri.23755

MAMON, R. S. and ELLIOTT, R. J., eds. (2014). *Hidden Markov Models in Finance: Further Developments and Applications, Volume II*. International Series in Operations Research &
*Management Science* **209**. Springer US, Boston, MA. https://doi.org/10.1007/978-1-4899-7442-6 MR3013963

MANN, H. B. and WALD, A. (1943). On the Statistical Treatment of Linear Stochastic Difference Equations. *Econometrica* **11** 173. https://doi.org/10.2307/1905674 MR0009291

NAKASHIMA, S., SUGHIYAMA, Y. and KOBAYASHI, T. J. (2020). Lineage EM Algorithm for Inferring Latent States from Cellular Lineage Trees. *Bioinformatics* **36** 2829–2838. https://doi.org/10.1093/bioinformatics/btaa040

OLARIU, V., COCA, D., BILLINGS, S. A., TONGE, P., GOKHALE, P., ANDREWS, P. W. and KADIRKAMANATHAN, V. (2009). Modified Variational Bayes EM Estimation of Hidden Markov Tree Model of Cell Lineages. *Bioinformatics* **25** 2824–2830. https://doi.org/10.1093/bioinformatics/btp456

RABINER, L. R. (1989). A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition. *Proceedings of the IEEE* **77** 257–286. https://doi.org/10.1109/5.18626

RASCH, D. and SCHOTT, D. (2018). *Mathematical Statistics*. John Wiley & Sons, Hoboken. MR3618868

ROMBERG, J., CHOI, H., BARANIUK, R. G. and KINGSBURY, N. (2000). Multiscale Classification Using Complex Wavelets and Hidden Markov Tree Models. In *Proceedings 2000 International Conference on Image Processing (Cat. No.00CH37101)* 371–374 vol.2. IEEE, Vancouver, BC, Canada. https://doi.org/10.1109/ICIP.2000.899396

SHAHDOOSTI, H. R. and HAZAVEI, S. M. (2017). Image Denoising in Dual Contourlet Domain Using Hidden Markov Tree Models. *Digital Signal Processing* **67** 17–29. https://doi.org/10.1016/j.dsp.2017.04.011

VAN DER VAART, A. W. (1998). *Asymptotic Statistics*. Cambridge Series in Statistical and Probabilistic Mathematics Cambridge University Press, Cambridge, UK; New York, NY, USA. https://doi.org/10.1017/CBO9780511802256 MR1652247

WALD, A. (1949). Note on the Consistency of the Maximum Likelihood Estimate. *The Annals of Mathematical Statistics* **20** 595–601. https://doi.org/10.1214/aoms/1177729952 MR0032169

WEIBEL, J. (2025). Ergodic Theorem for Branching Markov Chains Indexed by Trees with Arbitrary Shape. *Journal of Applied Probability*. https://doi.org/10.1017/jpr.2025.14

XIE, M., JIANG, Z. and SAINJU, A. M. (2018). Geographical Hidden Markov Tree for Flood Extent Mapping. In *Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining* 2545–2554. ACM, London United Kingdom. https://doi.org/10.1145/3219819.3220053

YU, D. and DENG, L. (2015). *Automatic Speech Recognition: A Deep Learning Approach*. Signals and Communication Technology. Springer London, London. https://doi.org/10.1007/978-1-4471-5779-3 MR3559045

ZUCCHINI, W. and MACDONALD, I. L. (2009). *Hidden Markov Models for Time Series: An Introduction Using R*, 0 ed. Chapman and Hall/CRC. https://doi.org/10.1201/9781420010893 MR2523850