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Asymptotic theory for time series containing missing and amplitude modulated observations. (English) Zbl 0531.62079
Let X(n) be generated by a stationary ergodic process and \(Y(n)=a(n)X(n)\) be observed, as when \(a(n)=0\) or 1 and there are missing values. The paper is concerned with the asymptotic properties of the autocovariances \[ C_ Y(\ell) = N^{-1} \sum^{N-\ell}_{1} Y(n)Y(n+\ell) \] or of the ratios \(C_ Y(\ell)/C_ a(\ell)\). A strong law of large numbers and a central limit theorem are proved under a variety of conditions, in the latter case being assumed that \[ X(n) = \sum^{\infty}_{0} \beta(j) \epsilon(n-j) \] where the \(\epsilon(n)\) are martingale differences. Other conditions are also imposed. The paper concludes with an analysis of the estimates of an autoregression obtained from the Yule-Walker equations when there are missing values.
Reviewer: E.J.Hannan

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M09 Non-Markovian processes: estimation
62G05 Nonparametric estimation