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Teoriya Veroyatnostei i ee Primeneniya, 1977, Volume 22, Issue 1, Pages 72–88 (Mi tvp3103)  

This article is cited in 55 scientific papers (total in 57 papers)

Limiting measures arising in the asymptotic theory of symmetric groups. I

A. M. Veršik, A. A. Šmidt

Leningrad
Abstract: We consider the joint distribution, with respect to the Haar measure, of a decreasing sequence of normalized lengths of cycles in a permutation, i. e. in an element of the symmetric group $S_n$ of degree $n$. We prove the existence of a limiting (as $n\to\infty$) distribution which is a measure in the space of non-negative series with unit sum. For this measure, it turns possible to find finite dimensional distributions and to study, in detail, its structure which is connected with some homogeneous Markov chain. This enables to obtain a large number of asymptotic formulas for invariant functionals on $S_n$, for example (main formula of Section 6),
$$ \lim_i\lim_n\frac{1}{n!}|\{g\in S_n:(\ln(n_i(g)/n)+i)/\sqrt i\le b\}|=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^b e^{-x^2/2}\,dx, $$
where $n_i(g)$ is the length of the $i$-th, in size, cycle in the permutation $g\in S_n$.
Received: 09.04.1975
English version:
Theory of Probability and its Applications, 1977, Volume 22, Issue 1, Pages 70–85
DOI: https://doi.org/10.1137/1122006
Bibliographic databases:
Language: Russian
Citation: A. M. Veršik, A. A. Šmidt, “Limiting measures arising in the asymptotic theory of symmetric groups. I”, Teor. Veroyatnost. i Primenen., 22:1 (1977), 72–88; Theory Probab. Appl., 22:1 (1977), 70–85
Citation in format AMSBIB
\Bibitem{VerShm77}
\by A.~M.~Ver{\v s}ik, A.~A.~{\v S}midt
\paper Limiting measures arising in the asymptotic theory of symmetric groups.~I
\jour Teor. Veroyatnost. i Primenen.
\yr 1977
\vol 22
\issue 1
\pages 72--88
\mathnet{http://mi.mathnet.ru/tvp3103}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=448476}
\zmath{https://zbmath.org/?q=an:0375.60007}
\transl
\jour Theory Probab. Appl.
\yr 1977
\vol 22
\issue 1
\pages 70--85
\crossref{https://doi.org/10.1137/1122006}
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    This publication is cited in the following 57 articles:
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