Moments of an exponential functional of random walks and 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 $μ_k = \ev(ξ^k) < 1$ 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.

8 pages

References in corpus (1)

• An exponential functional of random walks