Limit Theorems for Allocation of Particles over Different Cells with Restrictions to the Size of the Cells
A. N. Timashev · Theory of Probability and Its Applications · 2005
Equiprobable allocation schemes of n indistinguishable or distinguishable particles over N distinguishable cells are considered provided the fillings of the cells take on values in a fixed subset A of the set of nonnegative integers. Local normal and Poisson theorems are proved for the distributions of the number of cells, each of which contains exactly r particles, and for the number of cycles of length $r\in A$ in a permutation selected at random and equiprobable from the set of all permutations of order n with N cycles $(N\le n)$ whose lengths are elements of a set $A\subset{\bf N}$. It is assumed that $n,N\to\infty$ in the central domain.