Asymptotic behaviour of the first positions of uniform parking functions

TL;DR

This paper investigates the asymptotic distribution of initial positions in uniform parking functions, showing they become independent and uniform under certain conditions, with implications for related statistics.

Contribution

It provides new asymptotic results and bounds for the initial positions of uniform parking functions, extending previous work with improved convergence rates.

Findings

First positions are asymptotically i.i.d. and uniform under specified conditions

Established bounds for convergence rates of initial parking positions

Derived limit theorems for sums and maxima of initial parking positions

Abstract

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