On some local asymptotic properties of sequences with a random index

Abstract

We consider sequences of random variables with the index subordinated by a doubly stochastic Poisson process. A Poisson stochastic index process, or PSI-process for short, is a random process ψ(t) with the continuous time t which one can obtain via subordination of a sequence of random variables (ξj ), j = 0, 1, . . ., by a doubly stochastic Poisson process Π1(tλ) as follows: ψ(t) = ξΠ1(tλ) , t > 0. We suppose that the intensity λ is a nonnegative random variable independent of the standard Poisson process Π1. In the present paper we consider the case of independent identically distributed random variables (ξj ) with a finite variance. R. Wolpert and M. Taqqu (2005) introduce and investigate a type of the fractional Ornstein — Uhlenbeck (fOU) process. We provide a representation for such fOU process with the Hurst exponent H ∈ (0, 1/2) as a limit of scaled and normalized sums of independent identically distributed PSI-processes with an explicitly given intensity λ. This fOU process, locally at t = 0, approximates in the square mean the fractional Brownian motion with the same Hurst exponent H ∈ (0, 1/2). We examine in details two examples with the intensity corresponding to the R. Wolpert and M. Taqqu’s fOU process: a telegraph process, arising for ξ0 having the Rademacher distribution ±1 with probabilities 1/2, and a PSI-process with the uniform distribution for ξ0. For these two examples we derive exact and asymptotic formulae for a local modulus of continuity over a small time interval for a single PSI-process.