Infinite $F$-Divisibility of Integer-Valued Random Variables

Abstract

Consider $m \geqq 2$ nonnegative, integer-valued random variables $X_1^{(n)},\cdots, X_m^{(n)}$ satisfying $\sum^m_{j=1} X_j^{(n)} \leqq n$. If $(X_1^{(n)},\cdots, X_m^{(n)})$ is one member of a family of random vectors, indexed by different values of the bound $n$, Darroch has proposed a definition of "independence except for the constraint," termed $F$-independence, which relates members of the family through their conditional distributions. In this paper we study the limit theory for sums of nonnegative, integer-valued variables, when the sums are bounded and the variables $F$-independent. The $F$-independence analogue of infinite divisibility, termed infinite $F$-divisibility, is defined and characterized, and it is shown that limit distributions of sums of $F$-independent, asymptotically negligible variables are infinitely $F$-divisible. Conditions under which the limit is binomial are given. Our results apply to families of random variables, induced by the $F$-independence definition, and their role in the theory is discussed.