Abstract
In this paper, we study the long memory property of two processes based on the Ornstein-Uhlenbeck model. Their are extensions of the Ornstein-Uhlenbeck system for which in the classic version we replace the standard Brownian motion (or other Lacute{e}vy process) by long range dependent processes based on alpha -stable distribution. One way of characterizing long- and short-range dependence of second order processes is in terms of autocovariance function. However, for systems with infinite variance the classic measure is not defined, therefore there is a need to consider alternative measures on the basis of which the long range dependence can be recognized. In this paper, we study three alternative measures adequate for alpha -stable-based processes. We calculate them for examined processes and indicate their asymptotic behavior. We show that one of the analyzed Ornstein-Uhlenbeck process exhibits long memory property while the second does not. Moreover, we show the ratio of two introduced measures is limited which can be a starting point to introduction of a new estimation method of stability index for analyzed Ornstein-Uhlenbeck processes.
Highlights
The long range dependence phenomena, called long memory or the Joseph effect, first was introduced by Mandelbrot and Wallis in 1968 [1]
In this paper, we study the long memory property of two processes based on the Ornstein-Uhlenbeck model
The classic examples of long range dependent processes are Gaussian fractional integrated autoregressive moving average (FARIMA or ARFIMA) time series [5] and increments of fractional Brownian motion (FBM) [6], which is closely related to fractional Langevin equation motion [11] and is a generalization of the classic Brownian motion
Summary
The long range dependence phenomena, called long memory or the Joseph effect, first was introduced by Mandelbrot and Wallis in 1968 [1]. In contrast to the covariation which is defined only for processes based on symmetric aÀstable distribution, it is defined for general class of infinitely divisible processes [15, 25] This measure can be considered, similar as autocovariance in Gaussian case, as a tool of long range dependence recognition. The process given in (5) is a natural extension of the Ornstein-Uhlenbeck model driven by Levy process defined in (3) It is an extension of presented in [51] CAR(1) process with symmetric aÀstable Levy motion discussed in the context of the asymptotic behavior of the corresponding measures of dependence. 1⁄4bb eÀaðyÀuÞubÀ1du: As the second Ornstein-Uhlenbeck model based on long memory process, we consider some modification of the fractional stable noise used in 5, namely we analyse the process fYÃðtÞg given by: dYÃðtÞ þ aYÃðtÞdt 1⁄4 bdLÃa;jðtÞ ð9Þ with the noise term given by: LÃa;jðtÞ
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