Commun. Korean Math. Soc. 25 (2010), No. 1, pp. 119–128 DOI 10.4134/CKMS.2010.25.1.119 SOME RESULTS ON ASYMPTOTIC BEHAVIORS OF RANDOM SUMS OF INDEPENDENT IDENTICALLY DISTRIBUTED RANDOM VARIABLES Tran Loc Hung and Tran Thien Thanh Abstract. Let {X n ,n ≥ 1} be a sequence of independent identically distributed (i.i.d.) random variables (r.vs.), defined on a probability space (Ω, A,P ), and let {N n ,n ≥ 1} be a sequence of positive integer-valued r.vs., defined on the same probability space (Ω, A,P ). Furthermore, we assume that the r.vs. N n ,n ≥ 1 are independent of all r.vs. X n ,n ≥ 1. In present paper we are interested in asymptotic behaviors of the random sum S N n = X 1 + X 2 + ··· + X N n , S 0 =0, where the r.vs. Nn,n ≥ 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], [4], [2], [3], [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 ≥ 1 are assumed to follow concrete probability laws as Poisson, Bernoulli, binomial or geometry. 1. Introduction Let {X n ,n ≥ 1} be a sequence of independent identically distributed (i.i.d.) random variables (r.vs.), defined on a probability space (Ω, A,P ) and let {N n , n ≥ 1} be a sequence of positive integer-valued r.vs., defined on the same probability space (Ω, A,P ). Furthermore, we assume that the r.vs. N n ,n ≥ 1 are independent of all i.i.d.r.vs. X n ,n ≥ 1. From now on, the random sum is defined by (1) S N n = X 1 + X 2 + ··· + X N n , S 0 =0. Received October 16, 2008; Revised August 13, 2009. 2000 Mathematics Subject Classification. 60F05, 60G50. Key words and phrases. random sum, independent identically distributed random vari- ables, asymptotic behavior, Poisson law, Bernoulli law, binomial law, geometric law. c 2010 The Korean Mathematical Society 119