Iterated logarithm law for sample generalized partial autocorrelations

The so-called generalized partial autocorrelations for a regular stationary process xt are the array of real coefficients [phi][lambda],[lambda]([nu]) that are defined by the equation E((xt + [phi][lambda],1([nu])xt-1 + ... + [phi][lambda],[lambda]([nu])xt-[lambda])xt-v-j) = 0; J = 1, ..., [lambda]. If the xt process is an ARMA(p,q) and if the are usual estimates of [phi][lambda],j([nu]), such as the extended Yule-Walker estimates, then under the weak assumption that the noise in the xt process is a martingale difference sequence, an iterated logarithm law is obtained for (), which applied to the sample generalized partial autocorrelations , yields lim supn almost surely, for [lambda] [greater-or-equal, slanted] p + 1, where w(n) = (2n-1 log log n)1/2 and the constant K depends only on the MA parameters of the xt process. For stationary AR(p) models, the following finer result is also obtained: For [lambda] [greater-or-equal, slanted] p + 1, almost surely, lim .