Some limit results for Markov chains indexed by trees

Abstract

We consider a sequence of Markov chains $({\mathcal{X}}^n{)}_{n=1,2,...}$ with ${\mathcal{X}}^n=(X_{\sigma }^n{)}_{\sigma \in \mathcal{T}}$, indexed by the full binary tree $\mathcal{T}={\mathcal{T}}_0\cup {\mathcal{T}}_1\cup ...$, where ${\mathcal{T}}_k$ is the $k$th generation of $\mathcal{T}$. In addition, let $({\Sigma }_k{)}_{k=0,1,2,...}$ be a random walk on $\mathcal{T}$ with ${\Sigma }_k\in {\mathcal{T}}_k$ and ${\tilde{\mathcal{R}}}^n=({\tilde{R}}_t^n{)}_{t\ge 0}$ with ${\tilde{R}}_t^n:=X_{{\Sigma }_{[tn]}}$, arising by observing the Markov chain ${\mathcal{X}}^n$ along the random walk. We present a law of large numbers concerning the empirical measure process ${\tilde{\mathcal{Z}}}^n=({\tilde{Z}}_t^n{)}_{t\ge 0}$ where ${\tilde{Z}}_t^n={\sum }_{\sigma \in {\mathcal{T}}_{[tn]}}{\delta }_{X_{\sigma }^n}$ as $n\to \infty$. Precisely, we show that if ${\tilde{\mathcal{R}}}^n\to \mathcal{R}$ for some Feller process $\mathcal{R}=(R_t{)}_{t\ge 0}$ with deterministic initial condition, then ${\tilde{\mathcal{Z}}}^n\to \mathcal{Z}$ with $Z_t={\delta }_{\mathcal{L}(R_t)}$.

12 pages