The Annals of Probability

Random walks in cones

Denis Denisov and Vitali Wachtel

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We study the asymptotic behavior of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone. The main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments. For the proof of tail asymptotics and integral limit theorems, we use a strong approximation of random walks by Brownian motion. For the proof of local limit theorems, we suggest a rather simple approach, which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks. We also discuss some possible applications of our results to ordered random walks and lattice path enumeration.

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Ann. Probab. Volume 43, Number 3 (2015), 992-1044.

First available in Project Euclid: 5 May 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60F17: Functional limit theorems; invariance principles

Random walk exit time harmonic function Weyl chamber


Denisov, Denis; Wachtel, Vitali. Random walks in cones. Ann. Probab. 43 (2015), no. 3, 992--1044. doi:10.1214/13-AOP867.

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