Quantitative recurrence results for random walks

TL;DR

This paper establishes quantitative recurrence results for random walks in one and two dimensions through local almost sure central limit theorems, extending classical recurrence theorems with probabilistic precision.

Contribution

It introduces local almost sure central limit theorems for lattice and non-lattice random walks in one and two dimensions, providing quantitative recurrence insights.

Findings

Quantitative recurrence results for lattice random walks in the plane.

Local almost sure CLT for non-lattice random walks in line and plane.

Almost sure CLT for multidimensional non-lattice random walks.

Abstract

First, we prove a \emph{local almost sure central limit theorem} for lattice random walks in the plane. The corresponding version for random walks in the line was considered by the author in \cite{5}. This gives us a quantitative version of P\'olya's Recurrence Theorem \cite{6}. Second, we prove a \emph{local almost sure central limit theorem} for (not necessarly lattice) random walks in the line or in the plane, which will also give us quantitative recurrence results. Finally, we prove an \emph{almost sure central limit theorem} for multidimensional (not necessarly lattice) random walks. This is achieved by exploiting a technique developed by the author in \cite{5}.