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 (ξ_j), j = 0, 1, …, by a doubly stochastic Poisson process Π₁(tλ) via the substitution ψ(t) = \({{\xi }_{{{{\Pi }_{1}}(t\lambda )}}}\), t \( \geqslant \) 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 (ξ_j) 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.

中文翻译：

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关于随机索引序列的一些局部渐近性质

摘要

本文考虑由双重随机泊松过程控制的具有随机或随机指数的随机序列。泊松随机索引处理（PSI-处理）与连续时间ψ（一个随机过程吨由从属随机变量的序列（ξ获得）_Ĵ）Ĵ = 0，1，...，由双随机泊松过程Π ₁（吨λ）经由置换ψ（吨）= \（{{\ XI} _ {{{{\裨} _ {1}}（T \拉姆达）}}} \） ，吨 \（\ geqslant \ ） 0，其中假定独立于标准泊松过程的Π的随机强度λ ₁。在本文中，我们限制我们考虑独立同分布的随机变量（ξ的情况下_Ĵ）具有有限方差。我们发现与Hurst指数一个小数奥恩斯坦-Uhlenbeck过程的表示ħ中的无关的标准化总和相同的限制形式引入和由R.沃伯特和M. Taqqu（2005）研究了∈（0，1/2）具有明确给定的随机强度λ分布的分布式PSI过程。这个分数Ornstein–Uhlenbeck过程在相同的Hurst指数H下，在t = 0时提供了分数布朗运动的局部均方近似。∈（0，1/2）。我们将详细研究两个具有随机强度λ的PSI过程的示例，这些随机强度λ在Wolpert和Taqqu的意义上产生了分数阶的Ornstein–Uhlenbeck过程。这些是引起当ξ电报处理₀具有拉特马赫分布±1与概率1/2和与ξ的均匀分布的PSI-过程₀。对于这两个示例，我们为单个PSI过程在较小的固定时间范围内计算局部连续模数的精确值和渐近值。