Let

be a sequence of independent identically distributed (i.i.d.) random variables (r.vs.), defined on a probability space (

,A,P), and let

be a sequence of positive integer-valued r.vs., defined on the same probability space (

,A,P). Furthermore, we assume that the r.vs.

,

are independent of all r.vs.

,

. In present paper we are interested in asymptotic behaviors of the random sum

,

, where the r.vs.

,

obey some defined probability laws. Since the appearance of the Robbins's results in 1948 ([8]), the random sums

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

, in cases when the

,

are assumed to follow concrete probability laws as Poisson, Bernoulli, binomial or geometry.