Asymptotics for the partial autocorrelation function of a stationary process
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.
scientific article; zbMATH DE number 1546757
scientific article