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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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