Asymptotic behaviour of the first positions of uniform parking functions

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.