Total variation distance for Poisson subset numbers
Let \(A_0,A_1,\dots,A_k\) be subsets of \(\{1,2,\dots,n\}\) with fixed sizes \(a_0,a_1,\dots,a_k\) taken at random independently and uniformly from the all subsets with given sizes and let \(W=| \bigcap_{j=0}^k| \) be the size of their intersection. Explicit total variation bound is given between the distributions of \(W\) and the Poisson distribution with intensity \(\lambda_{k,n}=EW.\) The obtained bound allows the authors to show that the distribution of \(W\) converges to a Poisson distribution if \(a_i\to\infty\), \(i=0,1,\dots,k\), in such a way that \(\lambda_{k,n}\lambda_k\in (0,\infty).\) Reviewer's note: Considering \(\bar{A}_i\) instead of \(A_i\), \(i=0,1,\dots,k\) and treating the numbers \(n-a_0,n-a_1,\dots,n-a_k\) as the sizes of groups of particles, we arrive to the well-known problem of equiprobable allocation of the \(k+1\) groups particles of sizes \(n-a_0,n-a_1,\dots,n-a_k\) into \(n\) cells in such a way that each cell contains at most one particle of each group. In this case \(W\) is the number of empty cells after the allocation of all groups of particles. Under such interpretation Poisson and normal limit theorems for \(W\) (including estimates of the total variation distance) were obtained by the reviewer and \textit{V. G. Mikhailov} [Theory Probab. Appl. 27, 734--743 (1982); translation from Teor. Veroyatn. Primen. 27, No. 4, 684--692 (1982; Zbl 0517.60008)] in slightly different terms.
scientific article; zbMATH DE number 5082145