On the return to equilibrium (Q1848125)

The author considers a random walk on the group of integers starting from the origin and whose steps admit as possible values exactly two integers \(a \) and \(b\) such that \( a < 0 < b \). Then for the case \( a=-1\) the author gives an explicit expression for the law of the first return time to the origin \[ P(T = n(b+1))={b \over n(b+1)-1} \binom{n(b+1)}{nb} p^{nb} (1-p) ^n, \] where \(p=P(X=-1)\). The presentation is elegant and historical considerations are presented.