A Simple Analytic Proof of the Pollaczek-Wendel Identity for Ordered Partial Sums

Abstract

In this note we prove an identity due to Pollaczek (1952) and Wendel (1960). The identity describes the distributions of ordered partial sums of independent identically distributed random variables and thus generalizes Spitzer's identity. Our proof follows from a simple analytic argument applying a kind of Wiener-Hopf decomposition. We also give an extension of the Pollaczek-Wendel identity.