Random walks in cones

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.Comment: Published at http://dx.doi.org/10.1214/13-AOP867 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dyson Ferrari–Spohn diffusions and ordered walks under area tilts

“…The main difference with [6, Theorem 3] is that we find the limit for conditional distributions without determining the asymptotic behavior ofP r λ (τ > 1). (Recall once again that [6,Theorem 3] is proven under the assumption r = h λ a for some fixed a ∈ W .) Fix some ∈ (0, 1/2) and define the stopping time Proof.…”

“…The organization of these sections is described in Subsection 3.8. Many of our technical estimates rely on strong approximation techniques and on a refinement of recent results on random walks in Weyl chambers and on cones [6,8]. We can either think of L as being defined on L 2 (R + ) with zero boundary conditions at zero, or as being defined on L 2 (R).…”

Conditioned limit theorems for products of random matrices

“…Unfortunately, the WienerHopf factorization is not suited for studying the exit time probabilities for random walks in R d or walks based on dependent variables. Alternative approaches have been developed recently: we refer to [8,9,11,29]. For the results (1.4) and (1.5) we rely, on the one hand, upon these developments, and, on the other hand, upon a key strong approximation result for dependent random variables established separately in [17].…”

“…We refer the reader to Spitzer [28], Iglehart [20], Bolthausen [5], Bertoin and Doney [2], Doney [10], Borovkov [3,4], Vatutin and Wachtel [30], Caravenna [7] and to references therein. Random walks in R d conditioned to stay in a cone have been considered in Shimura [27], Garbit [14], Echelsbacher and König [11] and Denisov and Wachtel [8,9]. The case of Markov chains with bounded jumps were considered by Varapoulos [29] who obtained upper and lower bounds for the exit probability.…”

Conditioned limit theorems for products of positive random matrices

“…However, the studies on the subject of fluctuation was quite sparse a few years ago. Thanks to the approach of Denisov and Wachtel (2015) for random walks in Euclidean spaces and motivated by branching processes, I. Grama, E. Le Page and M. Peigné recently progressed for invertible matrices (Grama et al, 2014). Here we propose to develop the same strategy for matrices with positive entries by using Hennion and Hervé (2008).…”