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arXiv · 1008.1514

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

Tamas Szabados·Balazs Szekely
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Abstract

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.

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Tamas Szabados, Balazs Szekely. 2010-08-09. Moments of an exponential functional of random walks and permutations with given descent sets. https://doi.org/10.1023/b%3Amahu.0000040544.59987.08

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