A note on random permutations and extreme value distributions

Abstract

Let Ωn be the set of all permutations of the set Nn = {1, 2,..., n} and let us suppose that each permutation ω = (a1,..., an) ∈ n has probability 1/n!. For ω = (a1,..., an) let Xnj = |aj -aj+1|, j ∈ Nn, an+1 = a1, Mn = max{Xn1,...,Xnn}. We prove herein that the random variable Mn has asymptotically the Weibull distribution, and give some remarks on the domains of attraction of the Fréchet and Weibull extreme value distributions.