Asimptotic Normality in the Problem of Selfish Parking

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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 Xn 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 Xn. 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 Xn is investigated in this paper and the asymptotic normality for Xn is proved.
Translated title of the contributionАсимптотическая нормальность в задаче об эгоистичной парковке
Original languageEnglish
Pages (from-to)368-379
JournalVestnik St. Petersburg University: Mathematics
Issue number4
Early online date23 Dec 2019
StatePublished - Dec 2019

Scopus subject areas

  • Statistics and Probability


  • random filling
  • discrete parking problem
  • asymptotic behavior of moments and asymptotic normality


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