Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE RATES OF CONVERGENCE IN WEAK LIMIT THEOREMS FOR NORMALIZED GEOMETRIC SUMS

  • Hung, Tran Loc (Department of Mathematics and Statistics University of Finance and Marketing) ;
  • Kien, Phan Tri (Department of Mathematics and Statistics University of Finance and Marketing)
  • Received : 2019.08.15
  • Accepted : 2020.01.17
  • Published : 2020.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

The main purpose of this paper is to establish the rates of convergence in weak limit theorems for normalized geometric sums of independent identically distributed random variables via Zolotarev's probability metric.

Keywords

References

  1. S. Asmussen, Ruin Probabilities, Advanced Series on Statistical Science & Applied Probability, 2, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. https://doi.org/10.1142/9789812779311
  2. S. Asmussen, Applied Probability and Queues, second edition, Applications of Mathematics (New York), 51, Springer-Verlag, New York, 2003.
  3. S. G. Bobkov, Proximity of probability distributions in terms of Fourier-Stieltjes transforms, Russian Math. Surveys 71 (2016), no. 6, 1021-1079; translated from Uspekhi Mat. Nauk 71 (2016), no. 6(432), 37-98. https://doi.org/10.4213/rm9749
  4. J. L. Bon, Geometric sums in reliability evaluation of regenerative systems, Information Processes 2 (2002), no. 2, 161-163.
  5. B. V. Gnedenko and V. Yu. Korolev, Random Summation, CRC Press, Boca Raton, FL, 1996.
  6. J. Grandell, Risk theory and geometric sums, Information Processes 2 (2002), no. 2, 180-181.
  7. A. Gut, Probability: a graduate course, second edition, Springer Texts in Statistics, Springer, New York, 2013. https://doi.org/10.1007/978-1-4614-4708-5
  8. T. L. Hung, On a probability metric based on Trotter operator, Vietnam J. Math. 35 (2007), no. 1, 21-32.
  9. T. L. Hung, On the rate of convergence in limit theorems for geometric sums, Southeast-Asian J. of Sciences 2 (2013), no. 2, 117-130.
  10. V. Kalashnikov, Geometric sums: bounds for rare events with applications, Mathematics and its Applications, 413, Kluwer Academic Publishers Group, Dordrecht, 1997. https://doi.org/10.1007/978-94-017-1693-2
  11. L. B. Klebanov, Heavy Tailed Distributions, Research Gate, 2003.
  12. L. B. Klebanov, G. M. Maniya, and I. A. Melamed, A problem of V. M. Zolotarev and analogues of infinitely divisible and stable distributions in a scheme for summation of a random number of random variables, Teor. Veroyatnost. i Primenen. 29 (1984), no. 4, 757-760.
  13. V. Yu. Korolev and A. V. Dorofeeva, Bounds for the concentration functions of random sums under relaxed moment conditions, Theory Probab. Appl. 62 (2018), no. 1, 84-97; translated from Teor. Veroyatn. Primen. 62 (2017), no. 1, 104-121. https://doi.org/10.1137/S0040585X97T988502
  14. V. Yu. Korolev and A. V. Dorofeeva, Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions, Lith. Math. J. 57 (2017), no. 1, 38-58. https://doi.org/10.1007/s10986-017-9342-7
  15. V. Yu. Korolev and A. I. Zeifman, Convergence of statistics constructed from samples with random sizes to the Linnik and Mittag-Leffler distributions and their generalizations, J. Korean Statist. Soc. 46 (2017), no. 2, 161-181. https://doi.org/10.1016/j. jkss.2016.07.001
  16. V. Yu. Korolev, A. I. Zeifman, and A. Y. Korchagin, Asymmetric Linnik distributions as limit laws for random sums of independent random variables with finite variances, Inform. Primen. 10 (2016), no. 4, 21-33.
  17. S. Kotz, T. J. Kozubowski, and K. Podgorsky, The Laplace Distribution and Generalization, Springer Science + Business Media, LLC, 2001.
  18. T. J. Kozubowski, Geometric infinite divisibility, stability, and self-similarity: an overview, in Stability in probability, 39-65, Banach Center Publ., 90, Polish Acad. Sci. Inst. Math., Warsaw, 2010. https://doi.org/10.4064/bc90-0-3
  19. T. J. Kozubowski and K. Podgorski, Asymmetric Laplace distributions, Math. Sci. 25 (2000), no. 1, 37-46.
  20. V. M. Kruglov and V. Yu. Korolev, Limit theorems for random sums, Moskov. Gos. Univ., Moscow, 1990.
  21. W. Rudin, Principles of Mathematical Analysis, third edition, McGraw-Hill Book Co., New York, 1976.
  22. E. Sandhya and R. N. Pillai, On geometric infinite divisibility, J. Kerala Stat. Assoc. 10 (1999), 1-7.
  23. H. F. Trotter, An elementary proof of the central limit theorem, Arch. Math. 10 (1959), 226-234. https://doi.org/10.1007/BF01240790
  24. V. M. Zolotarev, Metric distances in spaces of random variables and of their distributions, Mat. Sb. (N.S.) 101(143) (1976), no. 3, 416-454, 456.
  25. V. M. Zolotarev, Approximation of the distributions of sums of independent random variables with values in infinite-dimensional spaces, Teor. Verojatnost. i Primenen. 21 (1976), no. 4, 741-758.
  26. V. M. Zolotarev, Probability metrics, Teor. Veroyatnost. i Primenen. 28 (1983), no. 2, 264-287.