Strong Law of Large Numbers for a Function of the Local Time of a Transient Random Walk on a Group

作者：Yinshan Chang, Qinwei Chen, Q. Meng, Xue Peng · 发表于：Journal of theoretical probability · 年份：2025 · DOI：10.1007/s10959-025-01464-3 · 被引用次数：2 · 研究领域：Mathematics

This paper presents the strong law of large numbers for a function of the local time of a transient random walk on a group, extending the research of Asymont and Korshunov (J Theoret Probab 33(4):2315–2336, 2020. https://doi.org/10.1007/s10959-019-00937-6) for random walks on the integer lattice Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^{d}$$\end{document}. Under some weaker conditions, we prove that a certain function of the local times converges almost surely and in L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{1}$$\end{document} and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document}. The proof is based mainly on the subadditive ergodic theorem.