A note on random permutations and extreme value distributions

Abstract

Let $\Omega_n$ be the set of all permutations of the set $N_n = \{1, 2, \dots, n\}$ and let us suppose that each permutation $\omega = (a_1, \dots, a_n) \in \Omega_n$ has probability $1/n!$. For $\omega = (a_1, \dots, a_n)$ let $X_{nj} = |a_j - a_{j+1}|$, $j \in N_n$, $a_{n+1} = a_1$, $M_n = \max\{X_{n1}, \dots, X_{nn}\}$. We prove herein that the random variable $M_n$ has asymptotically the Weibull distribution, and give some remarks on the domains of attraction of the Fréchet and Weibull extreme value distributions.