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

Let X₁, X₂, ⋯, Xₙ be n independent random variables each distributed uniformly over the interval (0, 1), and let Y₀, Y₁, ⋯, Yₙ be the respective lengths of the $n + 1$ segments into which the unit interval is divided by the ᵢ\. A fairly wide class of statistical problems is related to finding the distribution of certain functions of the Yⱼ; 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₌₀ hⱼ(Yⱼ) for quite arbitrary functions hⱼ(x), this result being essentially an extension of the classical integrals of Dirichlet. The cases of statistical interest correspond to hⱼ(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.