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

Title & Authors
SOME RESULTS ON ASYMPTOTIC BEHAVIORS OF RANDOM SUMS OF INDEPENDENT IDENTICALLY DISTRIBUTED RANDOM VARIABLES
Hung, Tran Loc; Thanh, Tran Thien;

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

Advanced Studies in Contemporary Mathematics, 2013. vol.23. 1, pp.171-180
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Central Limit Theorem for the Sum of a Random Number of Dependent Random Variables, Asian Journal of Mathematics & Statistics, 2011, 4, 3, 168
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On the rate of convergence in limit theorems for random sums via Trotter-distance, Journal of Inequalities and Applications, 2013, 2013, 1, 404
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An Estimate of the Probability Density Function of the Sum of a Random NumberNof Independent Random Variables, Journal of Computational Engineering, 2015, 2015, 1
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