On a Class of Problems Related to the Random Division of an Interval

Abstract

Let $X_1, X_2, \cdots, X_n$ be $n$ independent random variables each distributed uniformly over the interval (0, 1), and let $Y_0, Y_1, \cdots, Y_n$ be the respective lengths of the $n + 1$ segments into which the unit interval is divided by the $\{X_i\}$. A fairly wide class of statistical problems is related to finding the distribution of certain functions of the $Y_j$; these problems are reviewed in Section 1. The principal result of this paper is the development of a contour integral for the characteristic function (ch. fn.) of the random variable $W_n = \sum^n_{j=0} h_j(Y_j)$ for quite arbitrary functions $h_j(x)$, this result being essentially an extension of the classical integrals of Dirichlet. The cases of statistical interest correspond to $h_j(x) = h(x),$ independent of $j$. There is a fairly extensive literature devoted to studying the distributions for various functions $h(x)$. By applying our method these distributions and others are readily obtained, in a closed form in some instances, and generally in an asymptotic form by applying a steepest descent method to the contour integral.