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May 2015 Random walks in cones
Denis Denisov, Vitali Wachtel
Ann. Probab. 43(3): 992-1044 (May 2015). DOI: 10.1214/13-AOP867


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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Denis Denisov. Vitali Wachtel. "Random walks in cones." Ann. Probab. 43 (3) 992 - 1044, May 2015.


Published: May 2015
First available in Project Euclid: 5 May 2015

zbMATH: 1332.60066
MathSciNet: MR3342657
Digital Object Identifier: 10.1214/13-AOP867

Primary: 60G50
Secondary: 60F17 , 60G40

Keywords: Exit time , Harmonic function , Random walk , Weyl chamber

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 3 • May 2015
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