Asymptotic normality for a general statistic from a stationary sequence.(English)Zbl 0609.62025

Let $$\{Z_ i:$$ $$-\infty <i<+\infty \}$$ be a strictly stationary $$\alpha$$-mixing sequence. The author obtains a set of necessary and sufficient conditions for the joint asymptotic normality of a statistic $$t^ 0_{r_ n}(Z_ 1,...,Z_{r_ n})$$ based on $$Z_ 1,...,Z_{r_ n}$$ and the statistic $$t^{m_ n}_{s_ n}(Z_{s_ n+1},...,Z_{s_ n+m_ n})$$ based on $$Z_{s_ n+1},...,Z_{s_ n+m_ n}$$ where $$r_ n\geq m_ n+s_ n\geq s_ n\to \infty$$, $$s_ n/r_ n\to \rho^ 2$$ and $$t^ 0_ n(z_ 1,...,z_ n)$$ is a function from $${\mathbb{R}}^ n$$ to $${\mathbb{R}}$$. Results obtained extend earlier work of J. A. Hartigan [Ann. Stat. 3, 573-580 (1975; Zbl 0303.62015)] for i.i.d. sequences $$\{Z_ i\}$$.
Reviewer: B.L.S.Prakasa Rao

MSC:

 62E20 Asymptotic distribution theory in statistics 60G10 Stationary stochastic processes 60F05 Central limit and other weak theorems

Zbl 0303.62015
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