Limit theory for the sample covariance and correlation functions of moving averages. (English) Zbl 0605.62092

Davis, Richard; Resnick, Sidney

Ann. Stat. 14, 533-558 (1986).

The authors consider processes \(\{X_ t\}\) which may be represented as two-sided infinite moving averages of a sequence of i.i.d. random variates \(\{Z_ t\}\) which have regularly varying tail probabilities with index \(\alpha >0\). They derive the asymptotic distributional properties of the sample autocorrelation and autocovariance functions of \(\{X_ t\}\). In particular, in the infinite variance case \((0<\alpha <2)\), the sample autocorrelation is shown to be distributed asymptotically as the ratio of two independent stable random variables with indices \(\alpha\) /2 and \(\alpha\). Some applications to ARMA model identification and estimation using moment estimators are discussed.

Reviewer: E.McKenzie

MSC:

62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62E20	Asymptotic distribution theory in statistics
62M09	Non-Markovian processes: estimation

Keywords:

two-sided infinite moving averages; regularly varying tail probabilities; asymptotic distributional properties; autocovariance functions; infinite variance case; sample autocorrelation; stable random variables; ARMA model identification; moment estimators