On Convergence Of Random Walks Having Jumps With Finite Variances To Stable Levy Processes

Abstract

A functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes to Levy processes in the Skorokhod space. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Levy processes with mixed normal distributions, in particular, to stable Levy processes. Statistical analysis of the traffic in information flows in modern computational and telecommunication systems sometimes shows that this characteristics possesses the property of self-similarity. In applied probability this property is usually modeled by Levy processes. This communication gives some theoretical grounds to this convention. In (Kashcheev 2000, 2001) some functional limit theorems were proved for compound Cox processes with square integrable leading random measures. However, the class of limit processes for compound Cox processes having jumps with finite variances and such leading random measures cannot contain any stable Levy process besides the Wiener process. The aim of the present work is to fill this gap. Let D = D[0, 1] be a space of real-valued rightcontinuous functions defined on [0, 1] and having leftside limits. Let F be the class of strictly increasing continuous mappings of the interval [0, 1] onto itself. Let f be a non-decreasing function on [0, 1], f(0) = 0, f(1) = 1. Let ‖f‖ = sup s6=t ∣∣∣∣log f(t)− f(s) t− s ∣∣∣∣ . If ‖f‖ < ∞, then the function f is continuous and strictly increasing, hence, it belongs to F . Define the metric d0(x, y) in D[0, 1] as the greatest upper bound of positive numbers for whichF contains a function f such that ‖f‖ ≤ and