Electronic Communications in Probability

Some limit results for Markov chains indexed by trees

Abstract

We consider a sequence of Markov chains $(\mathcal X^n)_{n=1,2,...}$ with $\mathcal X^n = (X^n_\sigma)_{\sigma\in\mathcal T}$, 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 $\widetilde{\mathcal R}^n = (\widetilde R_t^n)_{t\geq 0}$ 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 $\widetilde{\mathcal R}^n \Rightarrow{n\to\infty} \mathcal R$ for some Feller process $\mathcal R = (R_t)_{t\geq 0}$ with deterministic initial condition, then $\widetilde{\mathcal Z}^n \Rightarrow{n\to\infty} \mathcal Z$ with $Z_t = \delta_{\mathcal L(R_t)}$.

Article information

Source
Electron. Commun. Probab., Volume 19 (2014), paper no. 77, 11 pp.

Dates
Accepted: 11 November 2014
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465316779

Digital Object Identifier
doi:10.1214/ECP.v19-3601

Mathematical Reviews number (MathSciNet)
MR3283608

Zentralblatt MATH identifier
1334.60036

Subjects
Primary: 60F15: Strong theorems
Secondary: 60F05: Central limit and other weak theorems

Rights

Citation

Czuppon, Peter; Pfaffelhuber, Peter. Some limit results for Markov chains indexed by trees. Electron. Commun. Probab. 19 (2014), paper no. 77, 11 pp. doi:10.1214/ECP.v19-3601. https://projecteuclid.org/euclid.ecp/1465316779

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