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Stochastic dominance of first return times for nearest-neighbor random walks on ℤd
Abstract
For a d-dimensional probability vector h=(h1,…, hd), let (Shn)n≥ 0 be a nearest-neighbor random walk on ℤd such that at each step, it moves to one of the two nearest neighbors in the i-th dimension with probability 1/2 hi (i=1,…, d). Let Th=inf{n≥ 1: Shn=(0,…,0)}, the first return time to the origin. For two d-dimensional probability vectors h' and h'' with the former majorizing the latter, we show that Th' is stochastically smaller than Th''. In particular, the first return time for the d-dimensional simple random walk is stochastically larger than Th for all d-dimensional probability vectors h.
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BibTeX
@misc{hsiau2026stochastic,
title = {{Stochastic dominance of first return times for nearest-neighbor random walks on $\mathbbZ^d$}},
author = {Shoou-Ren Hsiau and Ting-Yi Tsai and Yi-Ching Yao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36728},
eprint = {2609.36728},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36728},
}
RIS
TY - UNPB
TI - Stochastic dominance of first return times for nearest-neighbor random walks on ℤ^d
AU - Shoou-Ren Hsiau
AU - Ting-Yi Tsai
AU - Yi-Ching Yao
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36728
UR - https://arxiv.org/abs/2609.36728
AB - For a d-dimensional probability vector h=(h_1,…, h_d), let (S^h_n)_n≥ 0 be a nearest-neighbor random walk on ℤ^d such that at each step, it moves to one of the two nearest neighbors in the i-th dimension with probability 1/2 h_i (i=1,…, d). Let T^h=inf{n≥ 1: S^h_n=(0,…,0)}, the first return time to the origin. For two d-dimensional probability vectors h' and h'' with the former majorizing the latter, we show that T^h' is stochastically smaller than T^h''. In particular, the first return time for the d-dimensional simple random walk is stochastically larger than T^h for all d-dimensional probability vectors h.
ER -