On the exit time from an orthant for badly oriented random walks

Rodolphe Garbit (1)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

In a recent paper, K. Raschel and R. Garbit proved that the exponential decreasing rate of the probability that a random walk (with all exponential moments) stays in a $d$-dimensional orthant is given by the minimum on this orthant of the Laplace transform of the random walk increments, as soon as this minimum exists. In other cases, the random walk is ''badly oriented'', and the exponential rate may depend on the starting point $x$. We prove here that this rate is nevertheless asymptotically equal to the infimum of the Laplace transform, as some selected coordinates of $x$ tend to infinity.

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

Domaines: Probabilités [math.PR]

Rodolphe Garbit. On the exit time from an orthant for badly oriented random walks. 2015. ⟨hal-00963650v2⟩