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

TL;DR

This paper proves that the generating functions for survival and excursion probabilities of multidimensional random walks in convex cones are not rational, linking algebraic properties to asymptotic behaviors of these probabilities.

Contribution

It provides a new, elementary proof that the generating function of survival probabilities in a cone is not rational, complementing existing results on excursion probabilities.

Findings

The generating function for excursion probabilities is not rational.

The generating function for survival probabilities is not rational.

Rationality of these series is linked to their asymptotic behavior.

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($\tau$ > n), where $\tau$ is the first exit time from a given cone; the second series is that of the excursion probabilities P($\tau$ > 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.