On Some Local Asymptotic Properties of Sequences with a Random Index
- PDF / 360,449 Bytes
- 12 Pages / 612 x 792 pts (letter) Page_size
- 44 Downloads / 180 Views
EMATICS
On Some Local Asymptotic Properties of Sequences with a Random Index O. V. Rusakova,*, Yu. V. Yakubovicha,**, and B. A. Baevb,*** a
b
St. Petersburg State University, St. Petersburg, 199034 Russia National Research University Higher School of Economics, St. Petersburg, 190121 Russia * e-mail: [email protected] ** e-mail: [email protected] *** e-mail: [email protected] Received July 12, 2019; revised March 11, 2020; accepted March 19, 2020
Abstract—Random sequences with random or stochastic indices controlled by a doubly stochastic Poisson process are considered in this paper. A Poisson stochastic index process (PSI-process) is a random process with the continuous time ψ(t) obtained by subordinating a sequence of random variables (ξj), j = 0, 1, …, by a doubly stochastic Poisson process Π1(tλ) via the substitution ψ(t) = ξΠ1(t λ), t > 0, where the random intensity λ is assumed independent of the standard Poisson process Π1. In this paper, we restrict our consideration to the case of independent identically distributed random variables (ξj) with a finite variance. We find a representation of the fractional Ornstein–Uhlenbeck process with the Hurst exponent H ∈ (0, 1/2) introduced and investigated by R. Wolpert and M. Taqqu (2005) in the form of a limit of normalized sums of independent identically distributed PSI-processes with an explicitly given distribution of the random intensity λ. This fractional Ornstein–Uhlenbeck process provides a local, at t = 0, mean-square approximation of the fractional Brownian motion with the same Hurst exponent H ∈ (0, 1/2). We examine in detail two examples of PSI-processes with the random intensity λ generating the fractional Ornstein–Uhlenbeck process in the Wolpert and Taqqu sense. These are a telegraph process arising when ξ0 has a Rademacher distribution ±1 with the probability 1/2 and a PSI-process with the uniform distribution for ξ0. For these two examples, we calculate the exact and the asymptotic values of the local modulus of continuity for a single PSI-process over a small fixed time span. Keywords: fractional Ornstein–Uhlenbeck process, fractional Brownian motion, pseudo-Poisson process, random intensity, telegraph process, modulus of continuity. DOI: 10.1134/S1063454120030115
1. INTRODUCTION After the publication of Mandelbrot’s papers [1, 2], stochastic processes with continuous trajectories and self-similarity properties acquired not only a theoretical interest, but also important practical applications, especially in the fields of finance and telecommunications. The major process among self-similar stochastic processes is, certainly, the fractional Brownian motion (fBm), which is the Gaussian process with a zero initial value, stationary increments, and exponential growth of the variance t2H, where the time t > 0. Here H ∈ (0, 1] is the so-called Hurst exponent. In this paper, we study the fBm at H ∈ (0, 1/2) for a concave function of the accumulated variance. In this case, near zero, the fBm is locally approximated by the fractional Ornstein–Uhlenbeck
Data Loading...
