A Log Log Law for Maximal Uniform Spacings

Abstract

Let X1,X2,⋯X_1, X_2, \cdots be a sequence of independent uniformly distributed random variables on [0,1]\lbrack 0, 1\rbrack and KnK_n be the kkth largest spacing induced by X1,⋯,XnX_1, \cdots, X_n. We show that P(Kn≤(logn−log3n−log2)/nP(K_n \leq (\log n - \log_3n - \log 2)/n i.o.) = 1 where logj\log_j is the jj times iterated logarithm. This settles a question left open in Devroye (1981). Thus, we have liminf(nKn−logn+log3n)=−log2almost surely,\lim \inf(nK_n - \log n + \log_3n) = -\log 2 \text{almost surely}, and limsup(nKn−logn)/2log2n=1/kalmost surely\lim \sup(nK_n - \log n)/2 \log_2n = 1/k \text{almost surely}.