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Long memory time series models. (English) Zbl 0607.62111
The statistical analysis of many applied problems like an investigation of hydrological time series and others has led to the conclusion that the peak of the periodogram near the origin should be explained by a model with spectral density which is not bounded in the neighbourhood of zero frequency. For this reason the paper investigates time series models with a long memory, namely, a stationary process X(t) with covariance function \(R_ k,\sum | R_ k| =\infty\) is considered. As an example of such a process the paper investigates the fractionally differenced white noise \(X_ t\) satisfying \[ (1-B)^{\delta}X_ t=\epsilon_ t,\quad \delta \in (-1/2,1/2) \] where \(\epsilon_ t\) is a white noise, B is a back-shift operator \(BX_ t=X_{t-1}\); as well as the seasonal fractionally differenced white noise defined by \[ (1-B^ s)^{\delta}X_ t=\epsilon_ t \] or the seasonal persistent process \[ [(1-e^{i\omega}B)(1-e^{-i\omega}B)]^{\delta}X_ t=\epsilon_ t. \] Spectral densities, correlation functions, estimations of parameters are investigated.
Reviewer: I.G.Zhurbenko

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M15 Inference from stochastic processes and spectral analysis
62M09 Non-Markovian processes: estimation
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