On Asymptotic Normality in One Generalization of the Renyi Problem

Abstract

We consider a generalization of the well-known problem to randomly fill a long segment by unit intervals. On the segment [0, x], x ≥ 1, we place an open unit interval according to the F_x distribution law, which is the distribution of the left-hand endpoint of the unit interval, concentrated on the segment [0, x – 1]. Let the first allocated interval (t, t + 1) divide the segment [0, x] into two parts [0, t] and [t + 1, x] and they are filled independently of each other according to the following rules. On the segment [0, t] a point t₁ is selected randomly according to the law F_t and the interval (t₁, t₁ + 1) is placed. A point t₂ is selected randomly in the segment [t + 1, x] such that u = t₂ – t – 1 is a random variable distributed according to the law F_{x – t – 1}, and we place the interval (t₂, t₂ + 1). In the same way, the newly formed segments are then filled. If x < 1, then the filling process is considered to be complete and the unit interval is not placed on the segment [0, x]. At the end of the filling process, unit intervals are located on the segment [0, x] such that the distances between adjacent intervals are less than one. In this article, we consider distribution laws F_x with distribution densities such that their graphs are centrally symmetric with respect to the point (x – 1/2, 1/x – 1). In particular, this class of distributions includes the uniform distribution on the segment [0, x – 1] (the corresponding filling problem was previously investigated by other authors). Let N_x be the total amount of single units placed on the segment [0, x]. Our concern is the properties of the distribution of this random variable. We obtain an asymptotic description of the behavior of central moments and prove the asymptotic normality of the random variable N_x. In addition, we establish that the distributions of the random variables N_x are the same for all the distribution laws of the specified class.