Another Characterization of Long Memory Behavior
Different criteria exist to define long memory behavior. The two most used relate to the asymptotic decay of the autocovariance function of a process, and to the shape of its spectral density. In the case of a long memory process, the asymptotic decay of the autocovariance function is hyperbolic, while in the case of a short memory process, this decay is geometric. For the spectral density, while it is bounded for each frequency in the case of a short memory process, this density is unbounded at a finite number of frequencies in the case of a long memory process. A lot of processes exhibit such behavior and thus are considered as long memory processes. We can cite the FARIMA and the Gegenbauer process. There exists another way to characterize long memory behavior. It is possible to exhibit long memory behavior in terms of asymptotic behavior of the variance of the partial sum of a process. This variance is well known as the Allan variance (Allan, 1966). We can characterize long memory behavior of the FARIMA process using this criterion. However we show that it is not possible to characterize long memory behavior of the Gegenbauer process in this way. This is due to the shape of the spectral density of this kind of process. Indeed, the spectral density of a FARIMA process is unbounded at the zero frequency, while the Gegenbauer process's spectral density is unbounded at a non-zero frequency. It is the main reason why the Allan variance cannot characterize long memory behavior of the Gegenbauer process. We then introduce a new statistic called the generalized partial sum, which will permit generalization of the results obtained for the partial sum. We give the properties of this new statistic. Finally, we study the properties of the generalized partial sum for the class of Gegenbauer processes
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