Returns to the origin for random walks on \(\mathbb Z\) revisited (Q2501668)

This mathematical note considers symmetric random walks defined as follows: let \(X_k, k = 1, 2,\dots\), be independent and identically distributed random variables with \(P\{X_k = 1\} = P\{X_k = -1\} =\frac12\), and let \(S_m = \sum_{k=1}^m X_k\), with \(S_0 = 0\), be a simple random walk starting at 0. Relying on previous definitions and results, the present article examines random walks of even length \(m = 2n\) and the properties of the random variable \(R = R_n\) defined as the number of returns of the origin in a random walk of length \(2n\), i.e. \(R_n\) corresponds to the number of times it happens that \(S_i = 0\), for \(i = 1, 2, \dots,2n\). The author states the following explicit formula for computing factorial moments \(E[R_n^k]\), for even \(n\): \[ E[R_n^k]=k!\sum_{i=0}^k (-1)^{k-i} C_i^k C_n^{i/2+n}. \] Ordinary moments can be recovered from the formulae of \(E[R_n^k]\) on the basis of linear combinations with Stirling numbers of the second kind. The expansion formula for the asymptotic behaviour of \(E[R_n^k]\) is shown to be \(E[R_n^k]\sim (k!/(k/2)!)n^{k/2}\).