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 (ξ_j ), 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 non-negative 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 pr