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

摘要

In this article, we present large deviation results for a model {ζ_1+…+ζ_n:n≥ 1} which is close to a random walk. More precisely, we consider independent random variables {ζ_n:n≥ 1} such that {ζ_n:n≥ 2} are i.i.d. and a different distribution for ζ_1 is allowed. We prove large deviation estimates for P(NX < xT) and P(NX < oo) as x - oo, where Nx := inf [n > 1 : ζ_1 +… + ζ_n - x)- Moreover, we provide an asymptotically efficient simulation law for the estimation of P(NX < xT) and P(NX < oo) 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).