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Kenneth Alexander proved a uniform Law Of Iterated logarithm for Vapnik-Chervonenkis classes in the article Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm (Ann. Probab. volume 12, number 4 (1984), 1041-1067).

It states that given a VC class of sets $\cal C$ and a sequence of i.i.d. random variables $(Y_n)_n$, the following holds: $$ \limsup_{n \rightarrow \infty} \sup_{C \in \cal C} \frac{\left|\sum_{i=1}^{n} 1_{C}(Y_i) - nP_{Y_1}(C)\right|}{\sqrt{ 2 n \log \log n}} = \sup_{c \in \cal C} \left(P_{Y_1}(C)(1-P_{Y_1}(C))\right)^{1/2} $$

What if one has a slightly more general setting: the variables $(Y_n)_n$ 'live' on a general measure space $(\Omega, {\cal F}, \mu)$, where $\mu$ is a finite measure. Which conditions on the measure $\mu$ and the variables $(Y_n)_n$ should be imposed for the statement above to hold?

Best regards

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