NOTE ON AN ALMOST SURE INVARIANCE PRINCIPLE FOR SOME EMPIRICAL PROCESSES
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概要
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Let ＄￥{￥xi_{i}￥}＄ be a strictly stationary sequence of random variables which are distributed uniformly over the interval ＄[0,1]＄ and satisfy the strong mixing (s.m.) condition (1.1) ＄￥alpha(n)=￥sup_{-￥infty k+'￥iota}|P(A￥cap B)-P(A)P(B)|A￥in X^{k},Be-￥prime^{￥infty}￥downarrow 0＄ as ＄ n￥rightarrow￥infty＄ , where ＄￥vee J_{*}^{b}＄ is the a-algebra generated by ＄￥xi_{a},＄ ＄￥cdots,＄ ＄￥xi_{b}(￥alpha￥leqq b)＄ . Recently, Berkes and Philipp (1977) proved an almost sure invariance principle for some empirical processes by which a functional law of the iterated logarithm for the functions of s.m. sequences, a two-dimensional functional law of the iterated logarithm, etc., are easily obtained. In this note, we shall prove that Theorem 1 in Berkes and Philipp [1] remains true under the less restrictive s.m. condition.
- Yokohama City Universityの論文
- 1979-00-00
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