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<p>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. <a href="https://doi.org/10.1007/s10959-019-00937-6">https://doi.org/10.1007/s10959-019-00937-6</a>) for random walks on the integer lattice <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {Z}^{d}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>. Under some weaker conditions, we prove that a certain function of the local times converges almost surely and in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^{1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>. The proof is based mainly on the subadditive ergodic theorem.</p>

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

  • 期刊论文

  • 发表日期：

    2025年11月11日

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  • Yinshan Chang,
  • Qinwei Chen,
  • Qian Meng,
  • Xue Peng

摘要

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 \(\mathbb {Z}^{d}\) Z d . Under some weaker conditions, we prove that a certain function of the local times converges almost surely and in \(L^{1}\) L 1 and \(L^{2}\) L 2 . The proof is based mainly on the subadditive ergodic theorem.