Long-range Dependence trough Gamma-mixed Ornstein-Uhlenbeck Process

Abstract

The limit process of aggregational models---(i) sum of random coefficient AR(1) processes with independent Brownian motion (BM) inputs and (ii) sum of AR(1) processes with random coefficients of Gamma distribution and with input of common BM's,---proves to be Gaussian and stationary and its transfer function is the mixture of transfer functions of Ornstein--Uhlenbeck (OU) processes by Gamma distribution. It is called Gamma-mixed Ornstein--Uhlenbeck process ($\Gamma\mathsf{MOU}$). For independent Poisson alternating $0$-$1$ reward processes with proper random intensity it is shown that the standardized sum of the processes converges to the standardized $\Gamma\mathsf{MOU}$ process. The $\Gamma\mathsf{MOU}$ process has various interesting properties and it is a new candidate for the successful modelling of several Gaussian stationary data with long-range dependence. Possible applications and problems are also considered.