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Pré-Publication, Document De Travail Année : 2021

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

Résumé

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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Dates et versions

hal-03418842 , version 1 (08-11-2021)
hal-03418842 , version 2 (19-11-2021)

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