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Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series. (English) Zbl 0854.62084
Summary: We derive the asymptotic distributions of the sample mean, autocovariances, and autocorrelations for a time series whose autocovariance function $\{\gamma_k\}$ has the power law decay $\gamma_k \sim \lambda k^{- \alpha}$, $\lambda > 0$, $0 < \alpha < 1$, as $k \to \infty$. The results differ in important respects from the corresponding results for short-memory processes, whose autocovariance functions are absolutely summable. For long-memory processes the variances of the sample mean, and of the sample autocovariances and autocorrelations for $0 < \alpha \le 1/2$, are not of asymptotic order $n^{-1}$. When $0 < \alpha < 1/2$, the asymptotic distributions of the sample autocovariances and autocorrelations are not normal.

MSC:
62M10Time series, auto-correlation, regression, etc. (statistics)
62E20Asymptotic distribution theory in statistics
60F05Central limit and other weak theorems
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References:
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