Weak Convergence of Generalized Empirical Processes Relative to $d_q$ Under Strong Mixing

Abstract

Let $\{X_i: i \geqq 1\}$ be a strong-mixing sequence of uniform $\lbrack 0, 1 \rbrack$ rv's and $\{C_i\}$ a sequence of constants, and define the generalized empirical process by $U_N(t) = (\sum^N_{i=1} C_i^2)^{-\frac{1}{2}} \sum^N_{i=1} C_i(I_{\lbrack X_i\leqq t \rbrack} - t), 0 \leqq t \leqq 1$. In this paper, the weak convergence, relative to the Skorohod metric, of $(U_N/q)$ to a certain Gaussian process $(U_0/q)$ is proved under certain conditions on the constants $\{C_i\}$, the strong-mixing coefficient and the function $q$ defined on $\lbrack 0, 1 \rbrack$. The class of functions $q$ considered in this paper include those of the type $q(t) = \lbrack t(1 - t) \rbrack^\eta, \eta > 0$. The earlier results of Fears and Mehra [7] concerning empirical processes for $\phi$-mixing sequences are also improved by weakening the conditions on the $\phi$-mixing coefficient and the function $q$.