## 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 $$0 < \alpha \leq 1/2$$, 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:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62E20 Asymptotic distribution theory in statistics 60F05 Central limit and other weak theorems
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### References:

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