On the exit time from a cone for random walks with drift

Rodolphe Garbit (1) , Kilian Raschel (2)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
2 LMPT - Laboratoire de Mathématiques et Physique Théorique

We compute the exponential decay of the probability that a given multi-dimensional random walk stays in a convex cone up to time $n$, as $n$ goes to infinity. We show that the latter equals the minimum, on the dual cone, of the Laplace transform of the random walk increments. As an example, our results find applications in the counting of walks in orthants, a classical domain in enumerative combinatorics.

Mots clés: Random walk; Cones; Exit time; Laplace transform

Domaines: Probabilités [math.PR]; Combinatoire [math.CO]

Rodolphe Garbit, Kilian Raschel. On the exit time from a cone for random walks with drift. 2014. ⟨hal-00838721v4⟩