Large Deviation Results for Wave Governed Random Motions Driven by Semi-Markov Processes
- 1 June 2011
- journal article
- research article
- Published by Taylor & Francis Ltd in Communications in Statistics - Simulation and Computation
- Vol. 40 (9), 1342-1363
- https://doi.org/10.1080/03610918.2011.575501
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 Kac , M. ( 1974 ). A stochastic model related to the telegrapher's equation . Rocky Mountain Journal of Mathematics 4 : 497 – 509 . [Crossref] [Google Scholar] ; Orsingher, 1990 Orsingher , E. ( 1990 ). Probability law, flow function, maximum distribution of wave governed random motions and their connections with Kirchoff's laws . Stochastic Processes and their Applications 34 ( 1 ): 49 – 66 . [Google Scholar] ).Keywords
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