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Iterated logarithm law for sample generalized partial autocorrelations

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  • Truong-Van, B.

Abstract

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 .

Suggested Citation

  • Truong-Van, B., 1997. "Iterated logarithm law for sample generalized partial autocorrelations," Statistics & Probability Letters, Elsevier, vol. 33(2), pages 217-223, April.
    • Handle: RePEc:eee:stapro:v:33:y:1997:i:2:p:217-223
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    References listed on IDEAS

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    1. Truong-Van, B., 1995. "Invariance principles for semi-stationary sequence of linear processes and applications to ARMA process," Stochastic Processes and their Applications, Elsevier, vol. 58(1), pages 155-172, July.
    2. Chen, Zhao-Guo, 1990. "An extension of Lai and Wei's law of the iterated logarithm with applications to time series analysis and regression," Journal of Multivariate Analysis, Elsevier, vol. 32(1), pages 55-69, January.
    3. Lai, T. L. & Wei, C. Z., 1983. "Asymptotic properties of general autoregressive models and strong consistency of least-squares estimates of their parameters," Journal of Multivariate Analysis, Elsevier, vol. 13(1), pages 1-23, March.
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