Long-range Dependence trough Gamma-mixed Ornstein–Uhlenbeck Process
E. Iglói, L. Kossuth University
G. Terdik, L. Kossuth University
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 ($Gamma QTR{sf}{MOU}$QTR{sf}{)}.
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 $Gamma QTR{sf}{MOU}$ process. The $Gamma
QTR{sf}{MOU}$ 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.
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