Moments of an exponential functional of random walks and permutations with given descent sets |
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Authors: | Szabados Tamás Székely Balázs |
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Institution: | (1) Department of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3, H ép. V. em., H-1521 Budapest, Hungary;(2) Budapest University of Technology and Economics, Budapest |
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Abstract: | The exponential functional of simple, symmetric random walks with negative
drift is an infinite polynomial Y = 1 + ξ1 + ξ1ξ2 + ξ1ξ2ξ3 + ⋯ of independent
and identically distributed non-negative random variables. It has moments that are
rational functions of the variables μ
k
= E(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.
This revised version was published online in August 2006 with corrections to the Cover Date. |
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Keywords: | random walk exponential functional permutations with given descent sets Pascal's triangle infinite polynomials of random variables |
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