On asymptotic normality in one generalization of the Renyi problem

Abstract

In this present paper we consider one generalization of the well-known problem of random filling a segment of large length with unit intervals. On the segment [0, x], if x _ 1, in accordance with the distribution law Fx we will place an open interval of unit length. Fx is the distribution of the left end of the unit interval, which is concentrated on the segment [0, x − 1]. Let the first allocated interval take the place of (t, t + 1) and divide the segment [0, x] into two parts [0, t] and [t + 1, x], which are further filled independently of each other according to the following rules. On the segment [0, t] a point t1 is selected randomly in accordance with the law Ft, 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 Fx−t−1, and we place the interval (t2, t2 +1). In the same way then the newly formed segments are filled. If x < 1, then the filling process is considered complete and the unit interval is not placed on the segment [0, x]. At the end of the filling process, unit intervals will be located on the segment [0, x] so that the distances between adjacent intervals are less than one. In this article we consider distribution laws Fx having distribution densities such that their graphs have the property of central symmetry with respect to the point (x − 1/2, 1/x − 1). This class of distributions includes, in particular, a uniform distribution on the segment [0, x − 1], according to which the problem of random filling was investigated earlier by other authors. Let Nx be the total number of single units placed on the segment [0, x]. We are interested in 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 Nx. In addition, we establish that the distributions of the random variables Nx are the same for all the distribution laws of the specified class.