Large Deviation Results for Wave Governed Random Motions Driven by Semi-Markov Processes |
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Authors: | Claudio Macci |
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Institution: | 1. Dipartimento di Matematica , Università di Roma Tor Vergata , Rome , Italy macci@mat.uniroma2.it |
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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]). |
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Keywords: | Importance sampling Level crossing probability Telegrapher's process |
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