The Annals of Mathematical Statistics

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

D. A. Darling

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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.

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Ann. Math. Statist., Volume 24, Number 2 (1953), 239-253.

First available in Project Euclid: 28 April 2007

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Darling, D. A. On a Class of Problems Related to the Random Division of an Interval. Ann. Math. Statist. 24 (1953), no. 2, 239--253. doi:10.1214/aoms/1177729030.

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  • See Correction: D. A. Darling. Correction Notes: Correction to "On a class of Problems Related to the Random Divison of an Interval". Ann. Math. Statist., Volume 33, Number 2 (1962), 812--812.