Limit Theorem for the Middle Members of Ordered Cycle Lengths in Random $A$-Permutations
Teoriâ veroâtnostej i ee primeneniâ, Tome 54 (2009) no. 1, pp. 63-79
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In this article, random permutation $\tau_n$ is considered uniformly distributed on the set of all permutations with degree $n$ and with cycle lengths from fixed set $A$ (so-called $A$-permutations). Let $\zeta_n$ be the general number of cycles and $\eta_n(1)\leq\eta_n(2)\leq\cdots\leq\eta_n(\zeta_n)$ be the ordered cycle lengths in a random permutation $\tau_n$. The central limit theorem is obtained here for the middle members of this sequence, i.e., for random variables $\eta_n(m)$ with numbers $m=\alpha\log n+o(\sqrt{\log n})$ as $n\to\infty$ for fixed $\alpha\in(0,\sigma)$ and for some class of the sets $A$ with positive asymptotic density $\sigma$. The basic approach to the proof is the new three-dimensional Tauberian theorem. Asymptotic behavior of extreme left and extreme right members of this sequence was investigated earlier by the author.
Keywords:
random $A$-permutation, ordered cycle lengths of permutation, Tauberian theorem.
@article{TVP_2009_54_1_a3,
author = {A. L. Yakymiv},
title = {Limit {Theorem} for the {Middle} {Members} of {Ordered} {Cycle} {Lengths} in {Random} $A${-Permutations}},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {63--79},
publisher = {mathdoc},
volume = {54},
number = {1},
year = {2009},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2009_54_1_a3/}
}
TY - JOUR AU - A. L. Yakymiv TI - Limit Theorem for the Middle Members of Ordered Cycle Lengths in Random $A$-Permutations JO - Teoriâ veroâtnostej i ee primeneniâ PY - 2009 SP - 63 EP - 79 VL - 54 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TVP_2009_54_1_a3/ LA - ru ID - TVP_2009_54_1_a3 ER -
A. L. Yakymiv. Limit Theorem for the Middle Members of Ordered Cycle Lengths in Random $A$-Permutations. Teoriâ veroâtnostej i ee primeneniâ, Tome 54 (2009) no. 1, pp. 63-79. http://geodesic.mathdoc.fr/item/TVP_2009_54_1_a3/