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1999 Long-range Dependence trough Gamma-mixed Ornstein-Uhlenbeck Process
E. Igloi, G. Terdik
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Electron. J. Probab. 4: 1-33 (1999). DOI: 10.1214/EJP.v4-53


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.


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E. Igloi. G. Terdik. "Long-range Dependence trough Gamma-mixed Ornstein-Uhlenbeck Process." Electron. J. Probab. 4 1 - 33, 1999.


Accepted: 15 September 1999; Published: 1999
First available in Project Euclid: 4 March 2016

zbMATH: 0934.60030
MathSciNet: MR1713649
Digital Object Identifier: 10.1214/EJP.v4-53

Primary: 60H05
Secondary: 60G15 , 60G18 , 60H10

Keywords: Aggregational model , Fractional Brownianmotion input , heart rate variability , long-range dependence , Ornstein--Uhlenbeck process , ‎spectral representation , stationarity , Stochastic differentialequation

Vol.4 • 1999
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