A Log Log Law for Maximal Uniform Spacings

Abstract

Let $X_1, X_2, \cdots$ be a sequence of independent uniformly distributed random variables on $\lbrack 0, 1\rbrack$ and $K_n$ be the $k$th largest spacing induced by $X_1, \cdots, X_n$. We show that $P(K_n \leq (\log n - \log_3n - \log 2)/n$ i.o.) = 1 where $\log_j$ is the $j$ times iterated logarithm. This settles a question left open in Devroye (1981). Thus, we have $\lim \inf(nK_n - \log n + \log_3n) = -\log 2 \text{almost surely},$ and $\lim \sup(nK_n - \log n)/2 \log_2n = 1/k \text{almost surely}$.