Asymptotic normality for a general statistic from a stationary sequence (Q1086931)

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 \textit{J. A. Hartigan} [Ann. Stat. 3, 573-580 (1975; Zbl 0303.62015)] for i.i.d. sequences \(\{Z_ i\}\).