Paper detail

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

The exponential functional of simple, symmetric random walks with negative drift is an infinite polynomial $Y = 1 + ξ_1 + ξ_1 ξ_2 + ξ_1 ξ_2 ξ_3 + ...$ 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.