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 Tr. Mat. Inst. Steklova, 2013, Volume 282, Pages 315–335 (Mi tm3498)

Random $A$-permutations and Brownian motion

A. L. Yakymiv

Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia

Abstract: We consider a random permutation $\tau _n$ uniformly distributed over the set of all degree $n$ permutations whose cycle lengths belong to a fixed set $A$ (the so-called $A$-permutations). Let $X_n(t)$ be the number of cycles of the random permutation $\tau _n$ whose lengths are not greater than $n^t$, $t\in[0,1]$, and $l(t)=\sum_{i\leq t,i\in A}1/i$, $t>0$. In this paper, we show that the finite-dimensional distributions of the random process $\{Y_n(t)=(X_n(t)-l(n^t))/\sqrt{\varrho\ln n}$, $t\in[0,1]\}$ converge weakly as $n\to\infty$ to the finite-dimensional distributions of the standard Brownian motion $\{W(t),t\in[0,1]\}$ in a certain class of sets $A$ of positive asymptotic density $\varrho$.

DOI: https://doi.org/10.1134/S0371968513030217

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English version:
Proceedings of the Steklov Institute of Mathematics, 2013, 282, 298–318

Bibliographic databases:

UDC: 519.212.2+519.218.1

Citation: A. L. Yakymiv, “Random $A$-permutations and Brownian motion”, Branching processes, random walks, and related problems, Collected papers. Dedicated to the memory of Boris Aleksandrovich Sevastyanov, corresponding member of the Russian Academy of Sciences, Tr. Mat. Inst. Steklova, 282, MAIK Nauka/Interperiodica, Moscow, 2013, 315–335; Proc. Steklov Inst. Math., 282 (2013), 298–318

Citation in format AMSBIB
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