A note on random permutations and extreme value distributions (Q1869583)

Let \(\Omega_n\) be the set of all permutations of the set \([n]= \{1,2,\dots, n\}\), and suppose that each permutation \(\omega= (a_1,a_2,\dots, a_n)\in \Omega_n\) has the probability \(1/n\)!. For a permutation \(\omega\), let \(X_{nj}= |a_j- a_{j+1}|\), \(j\in [n]\), \(a_{n+1}= a_1\), and let also \(M_n= \max\{X_{n1},\dots, X_{nn}\}\). Then \(X_{n1},\dots, X_{nn}\) is a sequence of dependent random variables that satisfies the condition of strict stationarity, and the marginal distribution of the random variable \(X_{nj}\) is given by \({\mathbf P}(X_{nj}= k)= 2(n-k)/n(n- 1)\), with \(k\in [n- 1]= \{1,2,\dots, n-1\}\). The main result of this mathematical note is to prove the following Theorem. For every real number \(x\), the following equality holds: \[ \lim_{n\to\infty} {\mathbf P}\{M_n\leq x\sqrt{n}+ n\}= \begin{cases} e^{-x^2},\quad &\text{if }x< 0;\\ 1,\quad &\text{if }x\geq 0.\end{cases} \] This means that \(M_n\) has asymptotically the Weibull distribution function \(\Phi_\alpha(x)\), for \(\alpha= 2\). Several interesting consequences and remarks are given for the domains of attraction of the Fréchet, Weibull, and Gumbel extreme value distributions.