Asymptotic normality in the problem of selfish parking

Abstract

In the present work we continue to study one of the models of a discrete analogue of the Renyi problem, known as the “parking problem”. Let n, i be integers satisfying n ≥ 0 and 0 ≤ i ≤ n − 1. We place an open interval (i, i + 1) in the segment [0, n] with i being a random variable taking values 0, 1, 2, . . . , n − 1 with equal probability for all n ≥ 2. If n < 2 we say that the interval does not fit. After placing the first interval two free segments [0, i] and [i + 1, n] are formed and independently filled with the intervals of unit length according to the same rule, etc. At the end of the filling process the distance between any two adjacent unit intervals is at most 1. Let X_n denote the number of the unit intervals placed. In the previous work of the authors, published in 2018, the asymptotic behavior of the first moments of the random variable X_n was studied. In contrast to the classical case, the exact expressions were obtained for the expectation, variance, and third central moments. In this paper the asymptotic behavior of all central moments of the random variable X_n is studied and the asymptotic normality is proved for X_n.

Keywords:

random filling, discrete “parking” problem, asymptotic behavior of moments, asymptotic normality