Cutpoints and exchangeable events for random walks
Teoriâ veroâtnostej i ee primeneniâ, Tome 41 (1996) no. 4, pp. 854-868
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For a Markov chain $\{S_n\}$, call $S_k$ a cutpoint, and $K$ a cut-epoch, if there is no possible transition from $S_i$ to $S_j$ whenever $i$. We show that a transient random walk of bounded stepsize on an integer lattice has infinitely many cutpoints almost surely. For simple random walk on $\mathbf{Z}^d$, $d \ge 4$, this is due to Lawler. Furthermore, let $G$ be a finitely generated group of growth at least polynomial of degree 5; then for any symmetric random walk on $G$ such that the steps have a bounded support that generates $G$, the cut-epochs have positive density.
We also show that for any Markov chain which has infinitely many cutpoints almost surely, the eventual occupation numbers generate the exchangeable $\sigma$-field. Combining these results answers a question posed by Kaimanovich, and partially resolves a conjecture of Diaconis and Freedman.
Keywords:
cutpoint, exchangeable, Poisson boundary, random walks on groups.
Mots-clés : Markov chain
Mots-clés : Markov chain
@article{TVP_1996_41_4_a8,
author = {N. James and Y. Peres},
title = {Cutpoints and exchangeable events for random walks},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {854--868},
publisher = {mathdoc},
volume = {41},
number = {4},
year = {1996},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TVP_1996_41_4_a8/}
}
N. James; Y. Peres. Cutpoints and exchangeable events for random walks. Teoriâ veroâtnostej i ee primeneniâ, Tome 41 (1996) no. 4, pp. 854-868. http://geodesic.mathdoc.fr/item/TVP_1996_41_4_a8/