Tran Loc Hung and Tran Thien Thanh Hue University and Hue University

Abstract : Let $\{{X_{n},~ n\geq 1\}}$ be a sequence of independent identically distributed (i.i.d.) random variables (r.vs.), defined on a probability space $(\Omega, \mathcal{A}, P),$ and let $\{N_{n},~ n\geq 1\}$ be a sequence of positive integer-valued r.vs., defined on the same probability space $(\Omega, \mathcal{A}, P).$ Furthermore, we assume that the r.vs. $N_{n},~ n\geq 1$ are independent of all r.vs.~$X_{n},~ n\geq 1.$ In present paper we are interested in asymptotic behaviors of the random sum \[S_{N_{n}}=X_{1}+X_{2}+\cdots+X_{N_{n}}, \quad S_{0}=0,\] where the r.vs. $N_{n},~ n\geq 1$ obey some defined probability laws. Since the appearance of the Robbins's results in 1948 ([8]), the random sums $S_{N_{n}}$ have been investigated in the theory probability and stochastic processes for quite some time (see [1-5]). Recently, the random sum approach is used in some applied problems of stochastic processes, stochastic modeling, random walk, queue theory, theory of network or theory of estimation (see [10], [12]). The main aim of this paper is to establish some results related to the asymptotic behaviors of the random sum $S_{N_{n}},$ in cases when the $N_{n}, n\geq 1 $ are assumed to follow concrete probability laws as Poisson, Bernoulli, binomial or geometry.

Keywords : random sum, independent identically distributed random variables, random sum, independent identically distributed random variables, geometric law