Combinatorial Results in Fluctuation Theory

Abstract

The main purpose of the paper is to prove a combinatorial extension of a recent theorem of Baxter [6] concerning partial sums of random variables. The method of proof is closely related to that used by Sparre Andersen [1] in his basic 1949 paper which initiated the combinatorial approach to problems in fluctuation theory. The method does not seem to have been used since. The central idea behind Sparre Andersen's method, as it is used below, is conceptually very simple. It consists of verifying the validity of two operations on the finite sequences of real numbers under consideration. The first operation is referred to as "shrinking", and the second as "counting". In order to prove an invariant combinatorial result for such finite sequences of numbers, one shows first that if the result is true for a given sequence, it remains true as one decreases (or shrinks) the smallest number in the sequence (the shrinking Lemma 2.1) and, second, that if one inserts a sufficiently small number into a sequence for which the theorem holds, then the result also holds for the new sequence (the counting Lemma 2.2). Section 2 contains the fundamental combinatorial theorem concerning the joint behavior of the number of partial sums greater than zero and the number of them less than the last partial sum. Section 3 presents a probabilistic framework for the results of Section 2, as well as some further results.