Limiting Distributions of Homogeneous Functions of Sample Spacings

Abstract

Suppose $T_1, T_2, \cdots, T_n$ are the lengths of $n$ subintervals into which the interval $\lbrack 0, 1\rbrack$ is broken by $(n - 1)$ independent chance variables, each with a uniform distribution on $\lbrack 0, 1\rbrack$. Moran [1], Kimball [2], and Darling [3] have shown that if $r$ is a positive number, then the asymptotic distribution of $T_1^r + T_2^r + \cdots + T_n^r$ is normal. It is the purpose of this note to extend this result in two directions: more general functions of $T_1, \cdots, T_n$ are handled, and the joint distribution of several such functions is discussed. The proof is short and very simple.