SOME RESULTS ON ASYMPTOTIC BEHAVIORS OF RANDOM SUMS OF INDEPENDENT IDENTICALLY DISTRIBUTED RANDOM VARIABLES

Abstract

Let <TEX>${X_n,\;n\geq1}$</TEX> be a sequence of independent identically distributed (i.i.d.) random variables (r.vs.), defined on a probability space (<TEX>$\Omega$</TEX>,A,P), and let <TEX>${N_n,\;n\geq1}$</TEX> be a sequence of positive integer-valued r.vs., defined on the same probability space (<TEX>$\Omega$</TEX>,A,P). Furthermore, we assume that the r.vs. <TEX>$N_n$</TEX>, <TEX>$n\geq1$</TEX> are independent of all r.vs. <TEX>$X_n$</TEX>, <TEX>$n\geq1$</TEX>. In present paper we are interested in asymptotic behaviors of the random sum <TEX>$S_{N_n}=X_1+X_2+\cdots+X_{N_n}$</TEX>, <TEX>$S_0=0$</TEX>, where the r.vs. <TEX>$N_n$</TEX>, <TEX>$n\geq1$</TEX> obey some defined probability laws. Since the appearance of the Robbins's results in 1948 ([8]), the random sums <TEX>$S_{N_n}$</TEX> 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 <TEX>$S_{N_n}$</TEX>, in cases when the <TEX>$N_n$</TEX>, <TEX>$n\geq1$</TEX> are assumed to follow concrete probability laws as Poisson, Bernoulli, binomial or geometry.