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 (), the random sums
have been investigated in the theory probability and stochastic processes for quite some time (see , , , , ). 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 , ). 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.