PreprintCC BY-NC-ND 4.0arXiv
Geometric approximation for the number of returns of a transient Markov chain to its origin
Abstract
Given a discrete-time, transient Markov chain, we establish explicit total variation error bounds in the approximation of the number of visits to its starting state within the first n time steps by a geometric distribution. Our error bounds are expressed in terms of the heat kernel of the Markov chain. As applications, we consider Galton--Watson processes with a geometric offspring distribution, and random walks on the one- and three-dimensional integer lattices, regular trees, supercritical percolation clusters, and groups with polynomial growth.
Versions
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BibTeX
@misc{daly2026geometric,
title = {{Geometric approximation for the number of returns of a transient Markov chain to its origin}},
author = {Fraser Daly and Seva Shneer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.05039},
eprint = {2606.05039},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.05039},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Geometric approximation for the number of returns of a transient Markov chain to its origin
AU - Fraser Daly
AU - Seva Shneer
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2606.05039
UR - https://arxiv.org/abs/2606.05039
AB - Given a discrete-time, transient Markov chain, we establish explicit total variation error bounds in the approximation of the number of visits to its starting state within the first n time steps by a geometric distribution. Our error bounds are expressed in terms of the heat kernel of the Markov chain. As applications, we consider Galton--Watson processes with a geometric offspring distribution, and random walks on the one- and three-dimensional integer lattices, regular trees, supercritical percolation clusters, and groups with polynomial growth.
ER -