The Asymptotic Behavior of Spacings Under Kakutani's Model for Interval Subdivision

Abstract

If $X_1, X_2, \cdots$ are random variables with values in (0, 1), let $D_{n1}, \cdots, D_{n,n+1}$ denote the $n + 1$ spacings given by the first $n$ observations, $X_1, \cdots, X_n$. If $G^\ast_n$ denotes the empirical distribution function of the normalized spacings $\{(n + 1)D_{ni}\}$, it is proved in this paper that under the Kakutani model in which $X_m$ is a uniform random variable over the largest spacing determined by $X_1, \cdots, X_{m-1}$, with probability one $G^\ast_n \rightarrow G$ uniformly, where $G$ is the uniform distribution function on (0, 2). This is in sharp contrast to the known exponential limiting distribution when the $X_i$ are independent uniform random variables on (0, 1).