Sums of a Random Number of Random Variables

Abstract

We consider a family of random variables $\left\{ {\xi _i^n,\,i,\,n \in N} \right\},\,N = \left\{ {1,2, \ldots } \right\},$defined on a probability space {Ω, F, P} and a family $\left\{ {F_i^n,\,i \in {N_O},\,n \in N} \right\},\,{N_O} = \left\{ 0 \right\}U$of sub σ-algebras of F such that E ⁿ_i is F ⁿ_i - measurable and $F_i^n \subseteq F_{i + 1}^n,\,\,i \in {N_O},\,n \in N.$.Conditions for the convergence of the distribution of the sum

$${s_n} = \sum\limits_{i = 1}^{{k_n}} {\xi _i^n} $$

(1.1)

of independent random variables to the given infinitely divisible distribution are well known (see, for example, Petrov, 1987). Such conditions were considered in certain papers for a sum of dependent .Some conditions for the convergence to normal distribution have been studied by Brown (1971) (see also Dvoretzky, 1972). Analogical results were obtained for the case, when the infinitely divisible Unit distribution has a finite variance, by Brown and Eagleson (1971). Klopotowskl (1977) found such conditions without any assumption of the finiteness of moments.