On some local asymptotic properties of sequences with a random index

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 $\psi(t\lambda)$ with the continuous time $t$ which one can obtain via subordination of a sequence of random variables $(\xi_j)$, $j = 0, 1, \ldots$, by a doubly stochastic Poisson process $\Pi_1(t\lambda)$ as follows: $\psi(t) = \xi_{\Pi_1(t\lambda)}$, $t \geqslant 0$. We suppose that the intensity $\lambda$ is a nonnegative random variable independent of the standard Poisson process $\Pi_1$. In the present paper we consider the case of independent identically distributed random variables $(\xi_j)$ 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 $H \in (0, 1/2)$ as a limit of scaled and normalized sums of independent identically distributed PSI-processes with an explicitly given intensity $\lambda$. This fOU process, locally at $t = 0$, approximates in the square mean the fractional Brownian motion with the same Hurst exponent $H \in (0, 1/2)$. 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 $\xi_0$ having the Rademacher distribution $\pm1$ with probabilities $1/2$, and a PSI-process with the uniform distribution for $\xi_0$. 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.