A proof of Pollaczek-Spitzer identity (Q808104)

First, the author proves the following generalization of Takacs' ballot theorem. If \(r_ 1,r_ 2,...,r_ n\) are integers with sum \(s\geq 1\) and \(n_ r\) denotes the number of cyclic permutations in which all the partial sums are greater or equal to r with at least one sum equal to r, then \(\sum^{s}_{r=1}rn_ r=s\). Then this result is used to give a short (seemingly new) proof of the well-known Pollaczek-Spitzer identity.