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Abstract <p> A local limit theorem for random walks in the hyperbolic Poincaré space of dimension <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3045_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\ge 2\)</EquationSource> </InlineEquation> is proved. To this end, we use the model of a ball and describe the walk in it by means of the Möbius addition and multiplication. This also enables us to derive the corresponding law of large numbers. </p>

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Limit Theorems for Random Walks in the Hyperbolic Space

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • V. D. Konakov,
  • S. Menozzi

摘要

Abstract

A local limit theorem for random walks in the hyperbolic Poincaré space of dimension \(n\ge 2\) is proved. To this end, we use the model of a ball and describe the walk in it by means of the Möbius addition and multiplication. This also enables us to derive the corresponding law of large numbers.