On Sevast'yanov's theorem

In this paper we prove Poisson limit theorems for occupancy vectors Sn=X1n+⋯+Xnn (n⩾1), where {Xln:l=1,…,n} form some dependent k-dimensional random vectors with values in the set of linearly independent vectors e1,…,ek∈Rk. Our method extends the result of Sevast'yanov in [Sevast'yanov, B. A., 1972. Poisson limit law for a scheme of sums of dependent random variables. Theory Probab. Appl. 17, 695-699] for the case k=2.