Asymptotic properties of \(\omega^2\)-statistics from weakly dependent data (Q2742930)

Let \(X_1,X_2,\dotsc\) be a sequence of random variables which are considered as observations from random variables with distribution function \(F(x)\). Denote NEWLINE\[NEWLINE\omega^2(q)= n\int_{-\infty}^\infty \left(F_n(x)-F(x)\right)^2 q(F(x)) dF(x),NEWLINE\]NEWLINE where \(F_n(x)=n^{-1}\sum_{k=1}^n\delta_k(x)\) and NEWLINE\[NEWLINE\delta_k(x)=\begin{cases} 1,& X_k<x\\0,&X_k\geq x\end{cases}.NEWLINE\]NEWLINE Asymptotic properties of \(\omega^2\)-statistics for weakly dependent \(X_1,X_2, \dotsc\) are given. For the case of independent variables \(X_1,X_2,\dotsc\) see \textit{D.M. Chibisov}, Math. Stat. Probab. 6, 147-156 (1964).