Asymptotic Properties of MatricesRelated to Mappings of Partitions

Abstract

Let $S=(S_1\lz S_r)$ be a partition of the set $\cN=\{1\lz n\}$ into nonempty disjoint subsets, $\Phi$ a permutation on $\cN$, and $\xi_{ij}=|\Phi S_i\cap S_j|$ the cardinality of the intersection of the sets $\Phi S_i$ and $S_j$. Assuming that S is selected at random and equiprobably from the set of all the permutations satisfying the condition $|S_i|=s_i$, $i=1\lz r$, and the permutation $\Phi$ (possibly random) satisfies some constrains, local and integral limit theorems are proved for the joint distribution of the random variables $\xi_{ij}$, $i,j=1\lz r$, as $n\to\iy$ and $s_in^{-1}\lra a_j\in (0,1)$.