Limit Theorem for the Middle Members of Ordered Cycle Lengths in Random $A$-Permutations

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.