Large Deviation Results for Wave Governed Random Motions Driven by Semi-Markov Processes

Abstract

In this article, we present large deviation results for a model {ξ1 + … + ξ n : n ≥ 1} which is close to a random walk. More precisely, we consider independent random variables {ξ n : n ≥ 1} such that {ξ n : n ≥ 2} are i.i.d. and a different distribution for ξ1 is allowed. We prove large deviation estimates for P(N x ≤ xT) and P(N x < ∞) as x → ∞, where N x : = inf {n ≥ 1: ξ1 + … + ξ n ≥ x}. Moreover, we provide an asymptotically efficient simulation law for the estimation of P(N x ≤ xT) and P(N x < ∞) by Monte Carlo simulation based on the importance sampling technique. These results will be adapted to wave governed random motions driven by semi-Markov processes and we present some simulations. Finally, we study the convergence of some large deviation rates for standard wave governed random motions based on a scaling presented in the literature (see Kac, 1974; Orsingher, 1990).