SOME RESULTS ON ASYMPTOTIC BEHAVIORS OF RANDOM SUMS OF INDEPENDENT IDENTICALLY DISTRIBUTED RANDOM VARIABLES
0
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
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.
Abstract
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.
참고문헌 (14)
- W. Feller, "An Introduction to Probability Theory and Its Applications, volume II, 2nd edition" (1971), John Wiley & Sons
- B. Gnedenko, "Limit theorems for sums of a random number of positive independent random variables", Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, 통권 : II, pp.537~549 (1972), Univ. California Press
- B. Gnedenko, "On limit theorems for a random number of random variables, Probability theory and mathematical statistics", Lecture Notes in Math., 통권 : 1021, pp.167~176 (1983), Springer
- A. Krajka, "Necessary and sufficient conditions for weak convergence of random sums of independent random variables", Comment. Math. Univ. Carolinae, 34(3), pp.465~482 (1993)
- V. Kruglov, "Limit Theorems for Random Sums" (1990), Moscow University Press
- J. Melamed, "Limit theorems in the set-up of summation of a random number of independent identically distributed random variables", Lecture Notes in Math., 통권 : 1412, pp.194~228 (1989), Springer
- J. Modyorodi, "A central limit theorem for the sums of a random number of random variables", Ann. Univ. Sci. Budapest. Eotvos Sect. Math., 10, pp.171~182 (1967)
- H. Robbins, "The asymptotic distribution of the sum of a random number of random variables", Bull. Amer. Math. Soc., 54, pp.1151~1161 (1948)
- Z. Rychlik, "On the limit behaviour of sums of a random number of independent random variables", Colloq. Math., 28, pp.147~159 (1973)
- Z. Rychlik, "Convergence in law of random sums with nonrandom centering", J. Math. Sci., 106, pp.2860~2864 (2001)
- D. O. Selivanova, "Estimates for the rate of convergence in some limit theorems for geometric random sums", Moscow Univ. Comput. Math. Cybernet., (2), pp.27~31 (1995)
- Borkowski, Random sum estimators and their efficiency, Technical Report, Department of Mathematical Science, "Yu. B. Shvetlov", Montana State University, 1~20 (2004)
- D. Szasz, "Limit theorems for the distributions of the sums of a random number of random variables", Ann. Math. Statist., 43, pp.1902~1913 (1972)
- P. Vellaisamy, "Poisson and compound Poisson approximations for random sums of random variables", J. Appl. Probab., 33, pp.127~137 (1996)