BILLINGSLEY'S THEOREMS ON EMPIRICAL PROCESSES OF STRONG MIXING SEQUENCES

Let ＄￥{x_{j}, -￥infty<j<￥infty￥}＄ be a strictly stationary sequence of random variables satisfying some mixing condition with mixing coefficient ＄￥phi(n)＄ or ＄￥alpha(n)＄ . Let ＄F.(t)＄ be the empirical distribution function of ＄x_{1},￥cdots,＄ ＄x_{￥iota}＄ and ＄Y_{*}(, ￥omega)＄ ＄=n^{1/2}(F_{n}(t, ￥omega)-F(t))＄ . In [11, Billingsley proved the weak convergence theorem on ＄￥{Y_{n}￥}＄ under the condition ＄￥Sigma n^{2}￥phi^{1/2}(n)<￥infty＄ . (cf. Theorem 22.1 in [11). Recently, in [5], Sen proved the result under the condition ＄￥Sigma n￥phi^{1/2}(n)<￥infty＄ and in [61 Yokoyama proved it under the condition ＄￥Sigma ￥alpha^{￥beta}(n)<￥infty(0<￥beta<1/2)＄ . In this note, we shall show that Billingsley's theorem remains true under a less restrictive condition ＄￥alpha(n)=＄ ＄O(n^{-8-￥delta})(￥delta>0)＄ . A theorem corresponding to Theorem 22.2 in [11 is also proved (Section 4)