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

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Abstract : 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.
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Rodolphe Garbit, Kilian Raschel. On the exit time from a cone for random walks with drift. Revista Matemática Iberoamericana, European Mathematical Society, 2016, 32 (2), pp.511-532. ⟨10.4171/rmi/893⟩. ⟨hal-00838721v5⟩

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