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

  • Hung, Tran Loc (Department of Mathematics, Hue University) ;
  • Thanh, Tran Thien (Department of Mathematics, Hue University)
  • Published : 2010.01.31

Abstract

Let ${X_n,\;n\geq1}$ be a sequence of independent identically distributed (i.i.d.) random variables (r.vs.), defined on a probability space ($\Omega$,A,P), and let ${N_n,\;n\geq1}$ be a sequence of positive integer-valued r.vs., defined on the same probability space ($\Omega$,A,P). Furthermore, we assume that the r.vs. $N_n$, $n\geq1$ are independent of all r.vs. $X_n$, $n\geq1$. In present paper we are interested in asymptotic behaviors of the random sum $S_{N_n}=X_1+X_2+\cdots+X_{N_n}$, $S_0=0$, where the r.vs. $N_n$, $n\geq1$ 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 $N_n$, $n\geq1$ 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

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  1. Central Limit Theorem for the Sum of a Random Number of Dependent Random Variables vol.4, pp.3, 2011, https://doi.org/10.3923/ajms.2011.168.173
  2. On the rate of convergence in limit theorems for random sums via Trotter-distance vol.2013, pp.1, 2013, https://doi.org/10.1186/1029-242X-2013-404
  3. An Estimate of the Probability Density Function of the Sum of a Random NumberNof Independent Random Variables vol.2015, 2015, https://doi.org/10.1155/2015/801652