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

Abstract

Let X1,X2,⋯,XnX_1, X_2, \cdots, X_n be nn independent random variables each distributed uniformly over the interval (0, 1), and let Y0,Y1,⋯,YnY_0, Y_1, \cdots, Y_n be the respective lengths of the n+1n + 1 segments into which the unit interval is divided by the {Xi}\{X_i\}. A fairly wide class of statistical problems is related to finding the distribution of certain functions of the YjY_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 Wn=∑nj=0hj(Yj)W_n = \sum^n_{j=0} h_j(Y_j) for quite arbitrary functions hj(x)h_j(x), this result being essentially an extension of the classical integrals of Dirichlet. The cases of statistical interest correspond to hj(x)=h(x),h_j(x) = h(x), independent of jj. There is a fairly extensive literature devoted to studying the distributions for various functions h(x)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.