On the accuracy of the poissonisation in the infinite occupancy scheme (Q2234434)

The author considers a model with \({n\in\mathbb{N}}\) balls that are randomly placed in infinitely many cells numbered by integers. Let \(X_j\) be a place of the \(j\)-th ball and \({\mathbb{P}(X_j=i)=p_i>0}\) with \({\sum\limits_{i=1}^\infty p_i=1}, \ {p_i\geq p_{i+1}.}\) Let \({R^*_{n,k}}\) be the number of cells containing at least \({k\geq1}\) balls. For some Poisson flows \(P_i(t)\) with intensities \(p_i\) there is considered the random variable \({R^*_{P(t),k}=\sum\limits_{i=1}^\infty\mathbb{I}(P_i(t)\geq k)}.\) The goal of the paper is the statement on accuracy of a.s. approximation of \({R^*_{n,k}}\) by \({R^*_{P(n),k}}\) when \(n\) grows, for any fixed \({k\geq1},\) under condition of regular varying at infinity for \({\alpha(x)=\max\{j:\ p_j\geq1/x\}}.\)