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

Hung, Tran Loc ; Kien, Phan Tri ; Nhut, Nguyen Tan

Kybernetika, Tome 55 (2019) no. 6, pp. 961-975

Abstract (VO)

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.

Détail

DOI : 10.14736/kyb-2019-6-0961

Classification : 60E07, 60F05, 60F99, 60G50
Keywords: geometric random sums; Gnedenko's transfer theorem; Zolotarev probability metric

Bibliographie

[1] S., Asmusen,: Applied Probability and Queues. Springer, 2003. | DOI | MR

[2] S., Asmusen,: Ruin Probabilities. World Scientific, 2010. | MR

[3] G., Bobkov, S.: Proximity of probability distributions in terms of Fourier-Stieltjes transform. Russian Math. Surveys 71 (2016), 6, 1021-1079. | DOI | MR

[4] L., Bon, J.: Exponential Approximations of Geometric Sums for Reliability. University of Sciences at Technologies, 2002.

[5] L., Bon, J.: Geometric Sums in reliability evaluation of regenerative systems. In: Kalashnikov Memorial Seminar 2002, pp. 161-163. | MR

[6] M., Brown,: Error bounds for exponential approximation of geometric convolutions. Ann. Probab. 18 (1990), 3, 1388-1402. | DOI | MR

[7] F., Daly,: Compound geometric approximation under a failure rate constraint. J. Appl. Prob. 53 (2016), 700-714. | DOI | MR

[8] L., Dobrushin, R.: Lemma on the limit of compound random functions. Uspekhi Mat. Nauk 10 (1955), 2(64), 157-159. | DOI | MR

[9] W., Feller,: An Introduction to Probability Theory and its Applications. Volume 2. John Wiley and Son, Inc., 1966. | MR

[10] V., Gnedenko, B., G., Fahim,: On a transfer theorem. Dokl. Akad. Nauk SSSR 187 (1969), 1, 15-17. | MR

[11] V., Gnedenko, B.: On some stability theorems. Lect. Notes Math. 982 (1983), 24. | DOI | MR

[12] V., Gnedenko, B., Y., Korolev, V.: Random Summations: Limit Theorems and Applications. CRC Press, New York 1996. | MR

[13] L., Hung, T., T., L.,: On a Probability Metric Based on Trotter Operator. Vietnam Journal of Mathematics 35:1, (2007), 21-32. | MR

[14] L., Hung, T.: On the rate of convergence in limit theorems for geometric sums. Southeast-Asian J. Sci. 2 (2013), 2, 117-130.

[15] L., Hung, T., N., Hau, T.: On the accuracy of approximation of the distribution of negative-binomial random sums by Gamma distribution. Kybernetika 54 (2018), 5, 921-936. | DOI | MR

[16] V., Kalashnikov,: Geometric Sum: Bounds for Rare Events with Applications. Kluwer Academic Publishers, 1997. | DOI | MR

[17] B., Klebanov, L., M., Maniya, G., A., Melamed, I.: 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 | MR

[18] B., Klebanov, L.: Heavy Tailed Distributions. Research Gate, 2003.

[19] Y., Korolev, V., I., Zeifman, A.: A note on mixture representations for the Linnik and Mittag-Leffler distributions and their applications. J. Math. Sci. 218 (2016), 3, 314-327. | DOI | MR

[20] Y., Korolev, C., V., Dorofeeva, A.: Estimates for the concentration functions of random sums under weakened moment conditions (in Russian. Teor. Veroyatn. Primen. 62 (2017), 1, 104-121. | MR

[21] A., I., V. Y. Korolev, 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 | MR

[22] S., Kotz,, J., Kozubowski, T., K., Podgórsky,: The Laplace Distribution and Generalization. Springer Science and Business Media, LLC., 2001. | DOI | MR

[23] M., Kruglov, V., Yu., Korolev, V.: Limit Theorems for Random Sums. Moscow University Press, Moscow 1990. | MR

[24] Solym, Mawaki, Manou-Abi: Rate of convergence to alpha stable law using Zolotarev distance: technical report. Research Report, HAL Archives-Ouvertes 2017. | DOI

[25] E., Sandhya,, N., Pillai, R.: On geometric infinitely divisibility. J. Kerala Statist. Assoc. 10 (1999), 01-07.

[26] E., Sandhya,, N., Pillai, R.: Renewal theory and geometric infinite divisibility. ProbStat. Models 2 (2003), 1-8.

[27] V., Petrov, V.: Limit Theorems of Probability Theory (Sequences of Independent Random Variables). Clarendon Press Oxford, 1995. | DOI | MR

[28] W., Rudin,: Principles of Mathematical Analysis. Mc Graw-Hill, Inc. 1976. | DOI | MR

[29] M., Zolotarev, V.: Metric distances in spaces of random variables and their distributions. Mat. Sb. (N.S.) 101(143) (1976), 3, 416-454. | DOI | MR

[30] M., Zolotarev, V.: 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 | MR

[31] M., Zolotarev, V.: Probability metrics. Teor. Veroyatnost. i Primenen. 28 (1983), 2, 264-287. | MR