The limit law of certain discrete multivariate distributions

Let X₁,\,X₂,\,…,\,XN, N be independent but not necessarily identically distributed discrete and integer-valued random variables. Assume that X₁ m₁, X₂ m₂, …, XN mN almost surely, where m₁,\,m₂,…,\,mN are some integer numbers such that m₁+m₂+…+mN<0, and Xₖ is identically distributed as Xk+N, for all k in the sequence X₁,\,X₂,\,… In this communication, we make use of some of the known results to provide the closed-form expression of the limit multivariate distribution function P(X₁ x,\,X₁+X₂ x,\,…), x via: (1) inclusion-exclusion principle based product of the roots of GN(s)=1, where GN(s) is the probability generating function of SN=X₁+X₂+…+XN, (2) the probability mass function of SN, and (3) the expectation ESN.