On Limiting Distributions of a Random Number of Dependent Random Variables

Abstract

Let $\{X_n, n \geqq 1\}$ be a sequence of random variables such that for suitably chosen constants $a_n > 0$ and $b_n, n \geqq 1, \{(X_n - b_n)/a_n\}$ converges in distribution to a nondegenerate random variable $X$. Let $\{N_m, m \geqq 1\}$ be a sequence of positive, integer-valued random variables distributed independently of the sequence $\{X_n\}$ and converging to infinity in probability as $m\rightarrow \infty$. If $\{a_n\}$ and $\{b_n\}$ are the normalizing constants computed from a $\operatorname{cdf} F$ which is in the domain of attraction of one of the extreme value distributions and if the $\operatorname{cdf}$ of $X$ satisfies a condition determined by the domain of attraction to which $F$ belongs, then conditions on the limiting distribution of $\{N_m/m\}$ are obtained which are necessary and sufficient for the convergence in distribution of the sequence $\{(X_{N_m} - b_m)/a_m\}$ to a nondegenerate random variable $Y$. The $\operatorname{cdf}$ of $Y$ is either a location or a scale mixture of the $\operatorname{cdf}$ of $X$; and the $\operatorname{cdf} F$ is often unrelated to the distribution of $\{X_n\}$. These results extend a theorem stated by Berman; however, the method of proof is conceptually simpler.