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

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

2010-08-09

arxiv.org/abs/1008.1514v1

Periodica Mathematica Hungarica, Volume 49, Number 1 (2004)

Mathematics

Combinatorics

8 pages

Scientific paper

10.1023/B:MAHU.0000040544.59987.

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.

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