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The generating function of the survival probabilities in a cone is not rational

Abstract : A. We look at multidimensional random walks (Sn) n 0 in convex cones, and address the question of whether two naturally associated generating functions may define rational functions. The first series is the one of the survival probabilities P(τ > n), where τ is the first exit time from a given cone; the second series is that of the excursion probabilities P(τ > n, Sn = y). Our motivation to consider this question is twofold: first, it goes along with a global effort of the combinatorial community to classify the algebraic nature of the series counting random walks in cones; second, rationality questions of the generating functions are strongly associated with the asymptotic behaviors of the above probabilities, which have their own interest. Using wellknown relations between rationality of a series and possible asymptotics of its coefficients, recent probabilistic estimates immediately imply that the excursion generating function is not rational. Regarding the survival probabilities generating function, we propose a short, elementary and selfcontained proof that it cannot be rational neither.
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Preprints, Working Papers, ...
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Contributor : Rodolphe Garbit Connect in order to contact the contributor
Submitted on : Friday, November 19, 2021 - 10:07:09 AM
Last modification on : Saturday, November 20, 2021 - 3:49:14 AM


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  • HAL Id : hal-03418842, version 2
  • ARXIV : 2111.05027



Rodolphe Garbit, Kilian Raschel. The generating function of the survival probabilities in a cone is not rational. 2021. ⟨hal-03418842v2⟩



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