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Let {Xn, n≥1} be a sequence of independent identically distributed (i.i.d.) random variables (r.vs.), defined on a probability space (Ω,A, P), and let {Nn, n≥1} be a sequence of positive integer-valued r.vs., defined on the same probability space (Ω,A, P). Furthermore, we assume that the r.vs. Nn, n ≥1 are independent of all r.vs. Xn, n ≥1.
In present paper we are interested in asymptotic behaviors of the random sum SNn = X1 + X2 + · · · + XNn, S0 = 0,where the r.vs. Nn, n ≥ 1 obey some defined probability laws. Since the appearance of the Robbins’s results in 1948 ([8]), the random sums SNn 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 SNn, in cases when the Nn, n ≥ 1 are assumed to follow concrete probability laws as Poisson,Bernoulli, binomial or geometry.
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참고문헌 (14)
참고문헌 신청
A. Krajka / 1993 / Necessary and sufficient conditions for weak convergence of random sums of independent random variables / Comment. Math. Univ. Carolinae 34 (3) : 465 ~ 482
B. Gnedenko / 1972 / Limit theorems for sums of a random number of positive independent random variables / Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics an
B. Gnedenko / 1983 / On limit theorems for a random number of random variables, Probability theory and mathematical statistics / Lecture Notes in Math. : 167 ~ 176
D. O. Selivanova / 1995 / Estimates for the rate of convergence in some limit theorems for geometric random sums / Moscow Univ. Comput. Math. Cybernet. (2) : 27 ~ 31
D. Szasz / 1972 / Limit theorems for the distributions of the sums of a random number of random variables / Ann. Math. Statist. 43 : 1902 ~ 1913
B. Gnedenko / 1972 / Limit theorems for sums of a random number of positive independent random variables / Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics an
B. Gnedenko / 1983 / On limit theorems for a random number of random variables, Probability theory and mathematical statistics / Lecture Notes in Math. : 167 ~ 176
D. O. Selivanova / 1995 / Estimates for the rate of convergence in some limit theorems for geometric random sums / Moscow Univ. Comput. Math. Cybernet. (2) : 27 ~ 31
D. Szasz / 1972 / Limit theorems for the distributions of the sums of a random number of random variables / Ann. Math. Statist. 43 : 1902 ~ 1913
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