## A note on random permutations and extreme value distributions.(English)Zbl 1019.60050

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.

### MSC:

 60G70 Extreme value theory; extremal stochastic processes 05A05 Permutations, words, matrices
Full Text:

### References:

 [1] Gnedenko, B. V.: Sur la distribution limite du terme maximum d’une série aléatoire. Ann. Math., 44 , 423-453 (1943). · Zbl 0063.01643 · doi:10.2307/1968974 [2] de Haan, L.: On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Mathematical Centre Tracts 32, Mathematisch Centrum, Amsterdam (1970). · Zbl 0226.60039 [3] Kolchin, V. F.: Random Functions. Nauka, Moscow, (1984). (in Russian). [4] Leadbetter, M. R., Lindgren, G., and Rootzèn, H.: Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York-Heidelberg-Berlin (1983). · Zbl 0518.60021 [5] Mladenović, P: Limit theorems for the maximum terms of a sequence of random variables with marginal geometric distributions. Extremes, 2:4 , 405-419 (1999). · Zbl 0977.60021 · doi:10.1023/A:1009952232519
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.