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1 ULB - Université Libre de Bruxelles Bruxelles

Abstract : Random compositions of integers are used as theoretical models for many applications. The degree of distinctness of a composition is a natural and important parameter. A possible measure of distinctness is the number $X$ of distinct parts or components. This parameter has been analyzed in several papers. In this article we consider a variant of the distinctness: the number $Xm$ of distinct parts of multiplicity m that we call the $m$-distinctness. A firstmotivation is a question asked by Wilf for random compositions: what is the asymptotic value of the probability that a randomly chosen part size in a random composition of an integer $ν$ has multiplicity $m$. This is related to $\mathbb{E}Xm$, which has been analyzed by Hitczenko, Rousseau and Savage. Here, we investigate, from a probabilistic point of view, the first full part, the maximum part size and the distribution of $Xm$. We obtain asymptotically, as $ν → ∞$, the moments and an expression for a continuous distribution $φ$ , the discrete distribution of $Xm,ν $ being computable from $φ$ .

Keywords : generating functions saddle point method Mellin transforms urns models Poissonization





Autor: Guy Louchard -

Fuente: https://hal.archives-ouvertes.fr/



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