Moments of an exponential functional of random walks and permutations with given descent sets

TL;DR

This paper explores the moments of an exponential functional of symmetric random walks with negative drift, revealing connections to permutation enumeration with specified descent sets and providing recursive formulas for counting such permutations.

Contribution

It introduces a novel link between moments of exponential functionals and permutation descent set enumeration, along with a recursive method for counting permutations with given descent sets.

Findings

Moments are rational functions with universal coefficients.

Coefficients correspond to counts of permutations with specific descent sets.

A Pascal-type recursion enumerates permutations with given descent sets.

Abstract

The exponential functional of simple, symmetric random walks with negative drift is an infinite polynomial $Y = 1 + \xi_1 + \xi_1 \xi_2 + \xi_1 \xi_2 \xi_3 + ...$ of independent and identically distributed non-negative random variables. It has moments that are rational functions of the variables $\mu_k = \ev(\xi^k) < 1$ with universal coefficients. It turns out that such a coefficient is equal to the number of permutations with descent set defined by the multiindex of the coefficient. A recursion enumerates all numbers of permutations with given descent sets in the form of a Pascal-type triangle.