Asymptotic Normality in the Problem of Selfish Parking

Abstract

We continue in this paper to study one of the models of a discrete analog of the Renyi problem, also known as the parking problem. Suppose that n and i are 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, then 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 intervals of unit length according to the same rule, and so on. At the end of the process of filling the segment [0, n] with intervals of unit length, the distance between any two adjacent unit intervals does not exceed one. Suppose now that X_n is the number of unit intervals placed. In our earlier work published in 2018, we studied the asymptotic behavior of the first moments of random variable X_n. In contrast to the classical case, the exact expressions for the expectation, variance, and third central moment were obtained. The asymptotic behavior of all central moments of random variable X_n is investigated in this paper and the asymptotic normality for X_n is proved.