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.
Citation
Lionel Weiss. "Limiting Distributions of Homogeneous Functions of Sample Spacings." Ann. Math. Statist. 29 (1) 310 - 312, March, 1958. https://doi.org/10.1214/aoms/1177706734
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