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Some limit results for Markov chains indexed by trees


 
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1. Title Title of document Some limit results for Markov chains indexed by trees
 
2. Creator Author's name, affiliation, country Peter Czuppon; University of Freiburg; Germany
 
2. Creator Author's name, affiliation, country Peter Pfaffelhuber; University of Freiburg; Germany
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Tree-indexed Markov chain, weak convergence, tightness, random measure, empirical measure
 
3. Subject Subject classification 60F15; 60F05
 
4. Description 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 $Z_t = \delta_{\mathcal L(R_t)}$.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) DFG
 
7. Date (YYYY-MM-DD) 2014-11-11
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/3601
 
10. Identifier Digital Object Identifier 10.1214/ECP.v19-3601
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 19
 
12. Language English=en en
 
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