1. Subbotin, M. T. 1923. On the law of frequency of error. Mat. Sb. 31(2):296–301.
2. Box, G., and G. Tiao. 1973. Bayesian inference in statistical analysis. Reading, MA: Addison–Wesley. 608 p.
3. Evans, M., N. Hastings, and J.B. Peacock. 2000. Statistical distributions. 3rd. ed. N.Y.: John Wiley & Sons. 170 p.
4. Leemis, L.M., and J. T. McQueston. 2008. Univariate distribution relationships. Amer. Stat. 62(1):45–53.
5. RiskMetric Group, J. P.Morgan. 1996. RiskMetrics Technical Document. N.Y.
6. Varanasi, M.K., and B. Aazhang. 1989. Parametric generalized Gaussian density estimation. J. Acoust. Soc. Am. 86(4):1404–15.
7. Nadaraja, S. 2005. A generalized normal distribution. J. Appl. Stat. 32(7):685–94.
8. West,M. 1987. On scale mixtures of normal distributions. Biometrika 74(3):646–48.
9. Choy, S. T.B., andA. F. F. Smith. 1997. Hierarchical models with scale mixtures of normal distributions. Test 6:205– 21.
10. Zolotarev, V.M. 1983. Odnomernye ustoychivye raspredeleniya [One-dimensional stable distributions]. Moscow: Nauka. 304 p.
11. Korolev, V. Yu., V. E. Bening, L.M. Zaks, and A. I. Zeifman. 2012. Exponential power distributions as asymptotic approximations in applied probability and statistics. Applied Problems in Theory of Probabilities and Mathematical Statistics Related to Modeling of Information Systems (APTP + MS’2012). Book of abstracts of the 6th Workshop (International) (Autumn Session).Moscow: IPI RAN. 60– 71.
12. Barndorff-Nielsen, O. E. 1977. Exponentially decreasing distributions for the logarithm of particle size. Proc. R. Soc. Lond. Ser. A 353:401–19.
13. Barndorff-Nielsen,O.E., J.Kent, and S rensenM.. 1982. Normal variance-mean mixtures and z-distributions. Int. Stat. Rev. 50(2):145–59.
14. Korolev, V. Yu. 2013. Obobshhennye giperbolicheskie raspredelenija kak predel’nye dlja sluchajnyh summ[Generalized hyperbolic distributions as limit distributions for randomsums] [Theory Probab. Appl.] 58(1):117–32.
15. Zaks, L.M., and V. Yu. Korolev. 2013. Obobshchennye dispersionnye gamma-raspredelenija kak predel’nye dlja sluchajnyh summ [Variance-generalized-gammadistributions as limit laws for randomsums]. Inform. Appl. 7(1):105–15.
16. Gnedenko, B. V., and A.N. Kolmogorov. 1949. Predel’nye raspredelenija dlja summ nezavisimyh sluchajnyh velichin [Limit distributions for sums of independent random variables]. Moscow–Leningrad:GITTL. 264 p.
17. Gnedenko, B. V., and V. Yu. Korolev. 1996. Random summation: Limit theorems and applications. Boca Raton: CRC Press. 275 p.
18. Korolev, V. Yu. 1997. Postroenie modeley raspredeleniy birzhevykh tsen pri pomoshchi metodov asimptoticheskoy teorii sluchaynogo summirovaniya [Construction of models for stock prices by methods of the asymptotic theory of random summation]. Obozrenie Promyshlennoy i Prikladnoy Matematiki [Surveys in Applied and Industrial Mathematics] 4(1):86–102.
19. Korolev, V. Yu. 2000. Asimptoticheskie svoystva ekstremumov obobshchennyh protsessov Koksa i ikh primenenie k nekotorym zadacham finansovoy matematiki [Asymptotic properties of extrema of compound Cox processes and their applications to some problems of financial mathematics]. Theory Probab. Appl. 45(1):182–94.
20. Bening, V., and V.Korolev. 2002. Generalized Poisson models and their applications in insurance and finance. Utrecht: VSP. 434 p.
21. Korolev, V. Yu., and I. A. Sokolov. 2008. Matematicheskie modeli neodnorodnykh potokov ekstremal’nykh sobytiy [Mathematical models of nonhomogeneous flows of extremal events]. Moscow: TORUS PRESS. 200 p.
22. Korolev, V. Yu., V. E. Bening, and S. Ya. Shorgin. 2011. Matematicheskie osnovy teorii riska [Mathematical foundations of risk theory]. 2nd ed. Moscow: Fizmatlit.
23. Korolev, V. Yu. 2011. Veroyatnostno-statisticheskie metody dekompozitsii volatil’nosti khaoticheskikh processov [Probabilistic and statistical methods for the decomposition of the volatility of chaotic processes]. Moscow: Moscow University Press. 510 p.
24. Korolev, V., and N. Skvortsova, eds. 2006. Stochastic models of structural plasma turbulence. Utrecht: VSP. 400 p.
25. Korolev, V. Yu., I.G. Shevtsova, and S. Ya. Shorgin. 2011. O neravenstvakh tipa Berri–Jesseena dlya puassonovskikh sluchaynykh summ [On the Berry–Esseen-type inequalities forPoisson randomsums]. Informatics andApplications 5(3):64–66.
26. Korolev, V., and I. Shevtsova. 2012. An improvement of the Berry–Esseen inequality with applications to Poisson and mixed Poisson random sums. Scand. Actuar. J. 2:81–105. Available online since June 4, 2010. DOI:10.1080/03461238.2010.485370.

Title

ON CONVERGENCE OF THE DISTRIBUTIONS OF RANDOM SUMS TO SKEW EXPONENTIAL POWER LAWS

Journal

Informatics and Applications
December 2013, Volume 7, Issue 4, pp 66-74

Cover Date

2013-12-31

DOI

10.14357/19922264130407

Print ISSN

1992-2264

Publisher

Institute of Informatics Problems, Russian Academy of Sciences

Additional Links

• Editorial Board
• About This Journal
• Manuscript Submission

Key words

random sum; generalized Laplace distribution; skew generalized Laplace distribution; exponential power distribution; symmetric stable distribution; one-sided stable distribution; variance-mean normal mixture; mixed Poisson distribution; mixture of probability distributions; identifiable mixtures; additively closed family; convergence rate estimate

Authors

M. E. Grigor’eva and V.Yu. Korolev

Author Affiliations

   Parexel International, Moscow, Russia
   Faculty of Computational Mathematics and Cybernetics, M.V. Lomonosov Moscow State University; Institute of Informatics Problems, Russian Academy of Sciences, Moscow, Russia