Chen–Stein method for the uncovered set of random walk on $\mathbb {Z}_{n}^{d}$ for $d \ge 3$
| Autor: | Perla Sousi, Sam Thomas |
|---|---|
| Přispěvatelé: | Apollo - University of Cambridge Repository |
| Rok vydání: | 2020 |
| Předmět: |
Chen–Stein
Statistics and Probability N60G50 82C41A Probability (math.PR) late points 4901 Applied Mathematics uncovered set 60G50 60J10 82C41 Random walk random walk Combinatorics 4905 Statistics Total variation Cover (topology) Gaussian free field FOS: Mathematics 49 Mathematical Sciences 60J10 GFF Statistics Probability and Uncertainty Random variable Mathematics - Probability Mathematics |
| Zdroj: | Electron. Commun. Probab. |
| ISSN: | 1083-589X |
| DOI: | 10.1214/20-ecp331 |
| Popis: | Let $X$ be a simple random walk on $\mathbb{Z}_n^d$ with $d\geq 3$ and let $t_{\rm{cov}}$ be the expected cover time. We consider the set of points $\mathcal{U}_\alpha$ of $\mathbb{Z}_n^d$ that have not been visited by the walk by time $\alpha t_{\rm{cov}}$ for $\alpha\in (0,1)$. It was shown in [MS17] that there exists $\alpha_1(d)\in (0,1)$ such that for all $\alpha>\alpha_1(d)$ the total variation distance between the law of the set $\mathcal{U}_\alpha$ and an i.i.d. sequence of Bernoulli random variables indexed by $\mathbb{Z}_n^d$ with success probability $n^{-\alpha d}$ tends to $0$ as $n \to \infty$. In [MS17] the constant $\alpha_1(d)$ converges to $1$ as $d\to\infty$. In this short note using the Chen--Stein method and a concentration result for Markov chains of Lezaud we greatly simplify the proof of [MS17] and find a constant $\alpha_1(d)$ which converges to $3/4$ as $d\to\infty$. Comment: v3. Updated to author's new name, from "Thomas" to "Olesker-Taylor" |
| Databáze: | OpenAIRE |
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