A CLT When Summands Come Randomly from r Populations

Abstract

Let { X_n} be a sequence of mutually independent random variables (r.v.s.) defined on a probability space (Ω, β, P) with respective distribution functions (d.f.s.) {F_n }, all of which belong to the domain of normal attraction of a symmetric stable law with characteristic exponent a, 0 < a⩽2. Suppose further that at most r of the d.f.s. {F_n } are distinct, i.e. F_n ε {G₁, G₂,... G_r}.

A Central Limit Theorem type result is proved when the number of variables among {X₁ , X₂, ... , X_n} that follow a G_j is a r.v. satisfying certain conditions.