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Iterated logarithm law for sample generalized partial autocorrelations. (English) Zbl 0904.60023
Summary: The so-called generalized partial autocorrelations for a regular stationary process \(x_t\) are the array of real coefficients \(\varphi^{(\nu)}_{\lambda,\lambda}\) that are defined by the equation \[ E((x_t+ \varphi^{(\nu)}_{\lambda, 1}x_{t- 1}+\cdots+ \varphi^{(\nu)}_{\lambda, \lambda}x_{t-\lambda})x_{t-\nu-j})= 0;\qquad j= 1,\dots,\lambda. \] If the \(x_t\) process is an \(\text{ARMA}(p,q)\) and if the \(\widehat\varphi^{(\nu)}_{\lambda, j}\)’s are usual estimates of \(\varphi^{(\nu)}_{\lambda, j}\), such as the extended Yule-Walker estimates, then under the weak assumption that the noise in the \(x_t\) process is a martingale difference sequence, an iterated logarithm law is obtained for \((\widehat\varphi^{(q)}_{\lambda, j}- \varphi^{(q)}_{\lambda, j})\), which applied to the sample generalized partial autocorrelations \(\widehat\varphi^{(q)}_{\lambda, \lambda}\), yields \(\limsup_n w(n)^{-1}| \widehat\varphi^{(q)}_{\lambda, \lambda}|\leq K\) almost surely, for \(\lambda\geq p+1\), where \(w(n)= (2n^{-1}\log \log n)^{1/2}\) and the constant \(K\) depends only on the MA parameters of the \(x_t\) process. For stationary \(\text{AR}(p)\) models, the following finer result is also obtained: For \(\lambda\geq p+1\), almost surely, \(\limsup_n w(n)^{-1} \widehat\varphi^{(0)}_{\lambda, \lambda}= 1\) and \(\liminf_n w(n)^{-1} \widehat\varphi^{(0)}_{\lambda, \lambda}= -1\).
MSC:
60F15 Strong limit theorems
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62J05 Linear regression; mixed models
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