Asymptotic behaviour of the first positions of uniform parking functions

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