Strong Limiting Bounds for Maximal Uniform Spacings

Abstract

Let $U_1, U_2 \cdots$ be a sequence of independent uniformly distributed random variables on (0, 1) and $M_n$ be the largest spacing induced by $U_1, \cdots, U_n$. We show that $P(M_n \geq (\log n + 2 \log_2n + \log_3n + \cdots + \log_jn)/n \text{i.o.}) = 1$, where $\log_j$ is the $j$ times iterated logarithm, and $j \geq 4$. If $1 = N_1 < N_2 < \cdots < N_k < \cdots$ is the sequence of the successive times $n$ where $M_n < M_{n-1}$, we derive strong limiting bounds for $\{N_k, k \geq 1\}$.