The Law of the Iterated Logarithm for Empirical Processes
Part of the Progress in Probability book series (PRPR, volume 20)
Let (A,A,P) be a probability space and F ⊂ L2(A,A,P). Let X i , i ≥ 1, be a sequence of i.i.d. random variables with distribution P and let P n = n−1(δX 1 + ? + δX n ) be the n’th empirical measure for P. Using the methods employed to describe functional Donsker classes, we characterize when the normalized empirical process
satisfies the compact and bounded law of the iterated logarithm (LIL) uniformly over F. Sufficient conditions implying the bounded LIL are obtained. In particular, we obtain two new metric entropy integral conditions implying the bounded LIL. Moreover, the integral condition is essentially the best possible.
KeywordsCentral Limit Theorem Empirical Measure Empirical Process Iterate Logarithm Functional Limit Theorem
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