A Log Log Law for Maximal Uniform Spacings
نویسنده
چکیده
McGill University Let X1 , X2, • . . be a sequence of independent uniformly distributed random variables on [0, 1] and Kn be the kth largest spacing induced by X 1 , X12 . We show that P(Kn < (log n log3n log 2)/n i .o .) = 1 where log, is the j times iterated logarithm. This settles a question left open in Devroye (1981) . Thus, we have lim inf(nKn log n + log3n) _ -log 2 almost surely, and lim sup(nK, t log n)/2 log2n = 1/k almost surely .
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