Some limit results for Markov chains indexed by trees. (English) Zbl 1334.60036

Czuppon, Peter; Pfaffelhuber, Peter

Electron. Commun. Probab. 19, Paper No. 77, 11 p. (2014).

Summary: We consider a sequence of Markov chains \((\mathcal{X}^n)_{n=1,2,\dots{}}\) with \(\mathcal{X}^n = (X^n_{\sigma})_{\sigma\in\mathcal{T}}\), indexed by the full binary tree \(\mathcal{T}=\mathcal{T}_0 \cup \mathcal{T}_1 \cup \dots\), where \(\mathcal{T}_k\) is the \(k\)-th generation of \(\mathcal{T}\). In addition, let \((\Sigma_k)_{k=0,1,2,\dots{}}\) be a random walk on \(\mathcal{T}\) with \(\Sigma_k \in \mathcal{T}_k\) and \(\widetilde{\mathcal{R}}^n = (\widetilde{R}_t^n)_{t\geq 0}\) with \(\widetilde{R}_t^n := X_{\Sigma_{[tn]}}\), 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 \(\widetilde{\mathcal{Z}}^n = (\widetilde{Z}_t^n)_{t\geq 0}\), where \(\widetilde{Z}_t^n = \sum_{\sigma\in\mathcal{T}_{[tn]}} \delta_{X_{\sigma}^n}\) as \(n\to\infty\). Precisely, we show that, if \(\widetilde{\mathcal{R}}^n \Rightarrow \mathcal{R}\) for \(n\to \infty\) for some Feller process \(\mathcal{R}=(R_t)_{t\geq 0}\) with deterministic initial condition, then \(\widetilde{\mathcal{Z}}^n \Rightarrow \mathcal{Z}\) for \(n\to \infty\) with \(Z_t = \delta_{\mathcal{L}(R_t)}\).

MSC:

60F15	Strong limit theorems
60F05	Central limit and other weak theorems
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)
60G57	Random measures

Keywords:

tree-indexed Markov chain; law of large numbers; random measure; empirical measure; weak convergence; tightness