A limit theorem for the separable statistic in a random assignment scheme (Q1117553)

[For the entire collection see Zbl 0626.00025.] One considers the following scheme of random assignment of n particles in an infinite sequence of cells. Each particle is assigned to the k-th cell with probability \(p_ k\) \((\sum^{\infty}_{k=1}p_ k=1)\) and one assumes that \(p_ k\geq p_{k+1}\) and \(p_ k>0\) for each k. Let \(X_ k(n)\) be the number of particles in the k-th cell and let \(f_ 1(x)\), \(f_ 2(x),..\). be a sequence of real-valued functions defined for \(x=0,1,2,...\). Under certain conditions on the distribution of the probabilities \(\{p_ n,n\geq 1\}\) and on the sequence \(f_ 1(x),f_ 2(x),...\), one investigates the asymptotic normality of the random variable \(z(n)=\sum^{\infty}_{k=1}f_ k(X_ k(n))\). (The random variable \(Z_ n\) is proper since \(\sum^{\infty}_{k=1}X_ k(n)=n.)\)