Processes in random permutations

G. I. Ivchenko, Yu. I. Medvedev

Academy of Cryptography of the Russian Federation, Moscow

Abstract: The properties of the structure of random $n$-permutations are studied in terms of “tail” processes ${{C}_{n}}\left( b \right)=\sum\nolimits_{j>b}{{{c}_{j}}}$ – the number of cycles with lengths exceeding the level $b$, and ${{N}_{n}}\left( b \right)=\sum\nolimits_{j>b}{j{{c}_{j}}}$ – the number of elements contained in such cycles. The asymptotic behavior of these processes in the parametric model is described for the case when $b=\alpha n$, $0<\alpha <1$. A new three-parameter class of discrete distributions is obtained and a new class of chi-square statistical criteria is constructed for testing the hypothesis of the equiprobability of permutations and their effectiveness is investigated.

Key words: random permutations, parametric model, «tail» processes, discrete distributions, statistical inferences.

UDC: 519.212.2

Received 21.V.2025

DOI: 10.4213/mvk500