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Asymptotics for the partial autocorrelation function of a stationary process. (English) Zbl 0971.60039

Let \(\{X_t\}\) be a real discrete purely nondeterministic weakly stationary process with vanishing mean. Let \(\gamma(n)\) and \(\alpha(n)\) be its autocovariance function and partial autocorrelation function, respectively. The main result of the paper can be formulated as follows. If \(\gamma(n)\sim n^{2d-1}\ell(n)\) as \(n\to\infty\) where \(-\infty<d<\frac 12\) and \(\ell\) is a slowly varying function at infinity, then \(|\alpha(n)|\sim\gamma(n) /\sum_{k=-n}^n\gamma(k)\). In particular, if \(0<d<\frac 12\), then \(|\alpha(n)|\sim d/n\). The derivation of these formulas is based on asymptotic analysis of the relevant expected prediction error using precise asymptotics for the sequences of MA\((\infty)\) and AR\((\infty)\) coefficients of the process. The author proves a discrete-time analogue of the Seghier-Dym theorem concerning the intersection of past and future of a process. This result is used in the proof of the main theorem.
Reviewer: J.Anděl (Praha)

MSC:

60G10 Stationary stochastic processes
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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