Random walks in cones

Citation
Denis, Denisov et Vitali, Wachtel, Random walks in cones, Annals of probability (Online) , 43(3), 2015, pp. 992-1044
ISSN journal
2168894X
Volume
43
Issue
3
Year of publication
2015
Pages
992 - 1044
Database
ACNP
SICI code
Abstract
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