대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제33권2호
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  • Pages.605-618
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  • 2018
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  • 1225-1763(pISSN)
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  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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AN EXTENSION OF RANDOM SUMMATIONS OF INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM VARIABLES

  • 투고 : 2017.03.19
  • 심사 : 2018.02.01
  • 발행 : 2018.04.30
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초록

The main goal of this paper is to study an extension of random summations of independent and identically distributed random variables when the number of summands in random summation is a partial sum of n independent, identically distributed, non-negative integer-valued random variables. Some characterizations of random summations are considered. The central limit theorems and weak law of large numbers for extended random summations are established. Some weak limit theorems related to geometric random sums, binomial random sums and negative-binomial random sums are also investigated as asymptotic behaviors of extended random summations.

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과제정보

연구 과제 주관 기관 : Vietnam's National Foundation For Science and Technology Development (NAFOSTED)

참고문헌

  1. J. Blanchet and P. Glynn, Uniform renewal theory with applications to expansions of random geometric sums, Adv. in Appl. Probab. 39 (2007), no. 4, 1070-1097. https://doi.org/10.1239/aap/1198177240
  2. L. H. Y. Chen, L. Goldstein, and Q.-M. Shao, Normal approximation by Stein's method, Probability and its Applications (New York), Springer, Heidelberg, 2011.
  3. L. H. Y. Chen and Q.-M. Shao, Normal approximation for nonlinear statistics using a concentration inequality approach, Bernoulli 13 (2007), no. 2, 581-599. https://doi.org/10.3150/07-BEJ5164
  4. R. T. Durrett and S. I. Resnick, Weak convergence with random indices, Stochastic Processes Appl. 5 (1977), no. 3, 213-220. https://doi.org/10.1016/0304-4149(77)90031-X
  5. W. Feller, An Introduction to Probability Theory and its Applications. Vol. I, John Wiley and Sons, Inc., New York, 1957.
  6. B. V. Gnedenko, Limit theorems for sums of a random number of positive independent random variables, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, 537-549, Univ. California Press, Berkeley, CA, 1972.
  7. B. V. Gnedenko and V. Yu. Korolev, Random Summation, CRC Press, Boca Raton, FL, 1996.
  8. A. Gut, Probability: a graduate course, Springer Texts in Statistics, Springer, New York, 2005.
  9. C.-Y. Hu and T.-L. Cheng, A characterization of distributions by random summation, Taiwanese J. Math. 16 (2012), no. 4, 1245-1264. https://doi.org/10.11650/twjm/1500406734
  10. T. L. Hung and T. T. Thanh, Some results on asymptotic behaviors of random sums of independent identically distributed random variables, Commun. Korean Math. Soc. 25 (2010), no. 1, 119-128. https://doi.org/10.4134/CKMS.2010.25.1.119
  11. T. L. Hung, T. T. Thanh, and B. Q. Vu, Some results related to distribution functions of chi-square type random variables with random degrees of freedom, Bull. Korean Math. Soc. 45 (2008), no. 3, 509-522. https://doi.org/10.4134/BKMS.2008.45.3.509
  12. U. Islak, Asymptotic normality of random sums of m-dependent random variables, arXiv:1303.2386 [Math.PR], 2013.
  13. V. Kalashnikov, Geometric sums: bounds for rare events with applications, Mathematics and its Applications, 413, Kluwer Academic Publishers Group, Dordrecht, 1997.
  14. H. Klaver and N. Schmitz, An inequality for the asymmetry of distributions and a Berry- Esseen theorem for random summation, JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 1, Article 2, 12 pp.
  15. V. Yu. Korolev and V. M. Kruglov, Limit theorems for random sums of independent random variables, in Stability problems for stochastic models (Suzdal, 1991), 100-120, Lecture Notes in Math., 1546, Springer, Berlin, 1991.
  16. S. Kotz, T. J. Kozubowski, and K. Podgorski, The Laplace Distribution and Generalizations, Birkhauser Boston, Inc., Boston, MA, 2001.
  17. V. M. Kruglov and V. Yu. Korolev, Limit theorems for random sums, Moscow University Press, Moscow, (in Russian), 1990.
  18. J. A. Melamed, Limit theorems in the set-up of summation of a random number of independent identically distributed random variables, in Stability problems for stochastic models (Sukhumi, 1987), 194-228, Lecture Notes in Math., 1412, Springer, Berlin, 1989.
  19. E. Omey and R. Vesilo, Random sums of random variables and vectors: including infinite means and unequal length sums, Lith. Math. J. 55 (2015), no. 3, 433-450. https://doi.org/10.1007/s10986-015-9290-z
  20. A. Renyi, Probability theory, translated by LaszloVekerdi, North-Holland Publishing Co., Amsterdam, 1970.
  21. H. Robbins, The asymptotic distribution of the sum of a random number of random variables, Bull. Amer. Math. Soc. 54 (1948), 1151-1161. https://doi.org/10.1090/S0002-9904-1948-09142-X
  22. J. K. Sunklodas, On the rate of convergence of Lp norms in the CLT for Poisson and gamma random variables, Lith. Math. J. 49 (2009), no. 2, 216-221. https://doi.org/10.1007/s10986-009-9039-7
  23. J. K. Sunklodas, On the normal approximation of a binomial random sum, Lith. Math. J. 54 (2014), no. 3, 356-365. https://doi.org/10.1007/s10986-014-9248-6
  24. J. K. Sunklodas, On the normal approximation of a negative binomial random sum, Lith. Math. J. 55 (2015), no. 1, 150-158. https://doi.org/10.1007/s10986-015-9271-2
  25. P. Vellaisamy and B. Chaudhuri, Poisson and compound Poisson approximations for random sums of random variables, J. Appl. Probab. 33 (1996), no. 1, 127-137. https://doi.org/10.2307/3215270