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

Carlstein, Edward

Ann. Probab. 14, 1371-1379 (1986).

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

Keywords:

sample fractiles; general statistic; alpha-mixing sequence; necessary and sufficient conditions; joint asymptotic normality

Citations:

Zbl 0303.62015