The Stirling's formula and Polya's random walk theorem

This paper is about Stirling's formula and its application in random walk, The article first introduces the Stirling's formula and random walks, as well as their applications, and then provides proofs for the Stirling's formula and some lemmas. We have proved the Polya's random walk theorem using Stirling's’s formula and obtained the conclusion that in one and two dimensions, the points of the random walk will always return to the starting point, but in three and higher dimensions, the points of the random walk may not necessarily return to the starting point.