Polya's random walk theorem

TL;DR

This paper provides a proof of Pólya's random walk theorem utilizing classical techniques from special function theory and asymptotic analysis, offering a new perspective on a fundamental result in probability theory.

Contribution

It introduces a novel proof of Pólya's theorem based on classical special function and asymptotic analysis methods, differing from traditional probabilistic approaches.

Findings

Confirmed recurrence of 1D and 2D random walks

Established asymptotic behavior of return probabilities

Provided a classical analysis-based proof of Pólya's theorem

Abstract

This note presents a proof of P\'olya's random walk theorem using classical methods from special function theory and asymptotic analysis.