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

الأصل · EN

The exponential functional of simple, symmetric random walks with negative drift is an infinite polynomial Y = 1 + ξ₁ + ξ₁ ξ₂ + ξ₁ ξ₂ ξ₃ +... of independent and identically distributed non-negative random variables. It has moments that are rational functions of the variables μₖ = (ξᵏ) < 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.