The limit law of certain discrete multivariate distributions

Abstract

Let $X_1,X_2,\dots ,X_N$, $N\in \mathbb{N}$ be independent but not necessarily identically distributed discrete and integer-valued random variables. Assume that $X_1⩾m_1$, $X_2⩾m_2$, $\dots$, $X_N⩾m_N$ almost surely, where $m_1,m_2,\dots ,m_N$ are some integer numbers such that $m_1+m_2+\dots +m_N<0$, and $X_k$ is identically distributed as $X_{k+N}$, for all $k\in \mathbb{N}$ in the sequence $X_1,X_2,\dots$ In this communication, we make use of some of the known results to provide the closed-form expression of the limit multivariate distribution function $\mathbb{P}(X_1⩽x,X_1+X_2⩽x,\dots )$, $x\in \mathbb{Z}$ via: (1) inclusion-exclusion principle based product of the roots of $G_N(s)=1$, where $G_N(s)$ is the probability generating function of $S_N=X_1+X_2+\dots +X_N$, (2) the probability mass function of $S_N$, and (3) the expectation $\mathbb{E}{S}_N$.

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