Weak Convergence of a Two-Sample Empirical Process and a Chernoff-Savage Theorem for $|phi$-Mixing Sequences

Abstract

Using Pyke-Shorack (Ann. Math. Statist. (1968) 755-771) approach, based on weak-convergence properties of empirical processes, a Chernoff-Savage theorem concerning the asymptotic normality of two-sample linear rank statistics is proved for stationary $\phi$-mixing sequences $\{X_m\}$ and $\{Y_n\}$ of rv's. This main result (Theorem 4.1) is almost as strong as proved by Pyke and Shorack for sequences of independent rv's. The basic tool employed is the following new result concerning the behavior of empirical process $\{U_m(t): 0 \leqq t \leqq 1\}$ near 0 and 1 under $\phi$-mixing: For given $\varepsilon > 0$, the $P\lbrack(t(1 - t))^{-\frac{1}{2} + \delta}|U_m(t)| \leqq \varepsilon \forall 0 \leqq t \leqq \theta\rbrack, (0 < \delta < \frac{1}{2}, 0 < \theta < \frac{1}{2})$, can be made arbitrarily close to 1 by taking $m$ sufficiently large and $\theta$ sufficiently small.