Kybernetika 55 no. 6, 961-975, 2019

On asymptotic behaviors and convergence rates related to weak limiting distributions of geometric random sums

Tran Loc Hung, Phan Tri Kien and Nguyen Tan NhutDOI: 10.14736/kyb-2019-6-0961

Abstract:

Geometric random sums arise in various applied problems like physics, biology, economics, risk processes, stochastic finance, queuing theory, reliability models, regenerative models, etc. Their asymptotic behaviors with convergence rates become a big subject of interest. The main purpose of this paper is to study the asymptotic behaviors of normalized geometric random sums of independent and identically distributed random variables via Gnedenko's Transfer Theorem. Moreover, using the Zolotarev probability metric, the rates of convergence in some weak limit theorems for geometric random sums are estimated.

Keywords:

geometric random sums, Gnedenko's transfer theorem, Zolotarev probability metric

Classification:

60E07, 60F05, 60G50, 60F99

paper.pdf

References:

  1. S. Asmusen: Applied Probability and Queues. Springer, 2003.   DOI:10.1007/b97236
  2. S. Asmusen: Ruin Probabilities. World Scientific, 2010.   CrossRef
  3. S. G. Bobkov: Proximity of probability distributions in terms of Fourier-Stieltjes transform. Russian Math. Surveys 71 (2016), 6, 1021-1079.   DOI:10.1070/rm9749
  4. J. L. Bon: Exponential Approximations of Geometric Sums for Reliability. University of Sciences at Technologies, 2002.   CrossRef
  5. J. L. Bon: Geometric Sums in reliability evaluation of regenerative systems. In: Kalashnikov Memorial Seminar 2002, pp. 161-163.   CrossRef
  6. M. Brown: Error bounds for exponential approximation of geometric convolutions. Ann. Probab. 18 (1990), 3, 1388-1402.   DOI:10.1214/aop/1176990750
  7. F. Daly: Compound geometric approximation under a failure rate constraint. J. Appl. Prob. 53 (2016), 700-714.   DOI:10.1017/jpr.2016.35
  8. R. L. Dobrushin: Lemma on the limit of compound random functions. Uspekhi Mat. Nauk 10 (1955), 2(64), 157-159.   DOI:10.4213/rm9274
  9. W. Feller: An Introduction to Probability Theory and its Applications. Volume 2. John Wiley and Son, Inc., 1966.   CrossRef
  10. B. V. Gnedenko and G. Fahim: On a transfer theorem. Dokl. Akad. Nauk SSSR 187 (1969), 1, 15-17.   CrossRef
  11. B. V. Gnedenko: On some stability theorems. Lect. Notes Math. 982 (1983), 24.   DOI:10.1007/bfb0082058
  12. B. V. Gnedenko and V. Y. Korolev: Random Summations: Limit Theorems and Applications. CRC Press, New York 1996.   CrossRef
  13. T. L. Hung and T. L.: On a Probability Metric Based on Trotter Operator. Vietnam Journal of Mathematics 35:1, (2007), 21-32.   CrossRef
  14. T. L. Hung: On the rate of convergence in limit theorems for geometric sums. Southeast-Asian J. Sci. 2 (2013), 2, 117-130.   CrossRef
  15. T. L. Hung and T. N. Hau: On the accuracy of approximation of the distribution of negative-binomial random sums by Gamma distribution. Kybernetika 54 (2018), 5, 921-936.   DOI:10.14736/kyb-2018-5-0921
  16. V. Kalashnikov: Geometric Sum: Bounds for Rare Events with Applications. Kluwer Academic Publishers, 1997.   DOI:10.1007/978-94-017-1693-2
  17. L. B. Klebanov, G. M. Maniya and I. A. Melamed: A problem of Zolotarev and analogs of infinitely divisible and stable distributions in the scheme for summing a random number of random variables. Theory Probab. Appl. 29 (1084), 4, 791-794.   DOI:10.1137/1129104
  18. L. B. Klebanov: Heavy Tailed Distributions. Research Gate, 2003.   CrossRef
  19. V. Y. Korolev and A. I. Zeifman: A note on mixture representations for the Linnik and Mittag-Leffler distributions and their applications. J. Math. Sci. 218 (2016), 3, 314-327.   DOI:10.1007/s10958-016-3032-6
  20. C. Y. Korolev and A. V. Dorofeeva: Estimates for the concentration functions of random sums under weakened moment conditions (in Russian. Teor. Veroyatn. Primen. 62 (2017), 1, 104-121.   CrossRef
  21. V. Y. Korolev A. I. and 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), 2, 161-181.   DOI:10.1016/j.jkss.2016.07.001
  22. S. Kotz, T. J. Kozubowski and K. Podgórsky: The Laplace Distribution and Generalization. Springer Science and Business Media, LLC., 2001.   DOI:10.1007/978-1-4612-0173-1
  23. V. M. Kruglov and V. Yu. Korolev: Limit Theorems for Random Sums. Moscow University Press, Moscow 1990.   CrossRef
  24. Manou-Abi Solym Mawaki: Rate of convergence to alpha stable law using Zolotarev distance: technical report. Research Report, HAL Archives-Ouvertes 2017.   DOI:10.18642/jsata\_7100121914
  25. E. Sandhya and R. N. Pillai: On geometric infinitely divisibility. J. Kerala Statist. Assoc. 10 (1999), 01-07.   CrossRef
  26. E. Sandhya and R. N. Pillai: Renewal theory and geometric infinite divisibility. ProbStat. Models 2 (2003), 1-8.   CrossRef
  27. V. V. Petrov: Limit Theorems of Probability Theory (Sequences of Independent Random Variables). Clarendon Press Oxford, 1995.   DOI:10.1017/s001309150002335x
  28. W. Rudin: Principles of Mathematical Analysis. Mc Graw-Hill, Inc. 1976.   DOI:10.1017/s0013091500008889
  29. V. M. Zolotarev: Metric distances in spaces of random variables and their distributions. Mat. Sb. (N.S.) 101(143) (1976), 3, 416-454.   DOI:10.1070/sm1976v030n03abeh002280
  30. V. M. Zolotarev: Approximation of the distribution of sums of independent variables with values in infinite-dimensional spaces. Teor. Veroyatnost. i Primenen. 21 (1976), 4, 741-758.   DOI:10.1137/1121086
  31. V. M. Zolotarev: Probability metrics. Teor. Veroyatnost. i Primenen. 28 (1983), 2, 264-287.   CrossRef