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Asymptotic behaviour of the first positions of uniform parking functions

Part of: Probability theory on algebraic and topological structures Combinatorial probability Stochastic processes

Published online by Cambridge University Press:  20 April 2023

Etienne Bellin
Etienne Bellin*
Affiliation:
Ecole Polytechnique
*
*Postal address: Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, France. Email address: etienne.bellin@polytechnique.edu
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Abstract

In this paper we study the asymptotic behaviour of a random uniform parking function $\pi_n$ of size n. We show that the first $k_n$ places $\pi_n(1),\ldots,\pi_n(k_n)$ of $\pi_n$ are asymptotically independent and identically distributed (i.i.d.) and uniform on $\{1,2,\ldots,n\}$, for the total variation distance when $k_n = {\rm{o}}(\sqrt{n})$, and for the Kolmogorov distance when $k_n={\rm{o}}(n)$, improving results of Diaconis and Hicks. Moreover, we give bounds for the rate of convergence, as well as limit theorems for certain statistics such as the sum or the maximum of the first $k_n$ parking places. The main tool is a reformulation using conditioned random walks.

Keywords

Random walkstotal variation distanceCayley trees

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60G50: Sums of independent random variables; random walks 60B10: Convergence of probability measures
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 4 , December 2023 , pp. 1201 - 1218
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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