Informatics and Applications
2016, Volume 10, Issue 4, pp 2133
ASYMMETRIC LINNIK DISTRIBUTIONS AS LIMIT LAWS FOR RANDOM SUMS OF INDEPENDENT RANDOM VARIABLES WITH FINITE VARIANCES
 V. Yu. Korolev
 A. I. Zeifman
 A. Yu. Korchagin
Abstract
Linnik distributions (symmetric geometrically stable distributions) are widely applied in financial mathematics, telecommunication systems modeling, astrophysics, and genetics. These distributions are limiting for geometric sums of independent identically distributed random variables whose distribution belongs to the domain of normal attraction of a symmetric strictly stable distribution. In the paper, three asymmetric generalizations of the Linnik distribution are considered. The traditional (and formal) approach to the asymmetric generalization of the Linnik distribution consists in the consideration of geometric sums of random summands whose distributions are attracted to an asymmetric strictly stable distribution. The variances of such summands are infinite. Since in modeling real phenomena, as a rule, there are no solid reasons to reject the assumption of the finiteness of the variances of elementary summands, in the paper, two alternative asymmetric generalizations are proposed based on the representability of the Linnik distribution as a scale mixture of normal laws or a scale mixture of Laplace laws. Examples are presented of limit theorems for sums of a random number of independent random variables with finite variances in which the proposed asymmetric Linnik distributions appear as limit laws.
[+] References (54)
 Mittnik, S., and S. Rachev. 1993. Modeling asset returns with alternative stable models. Economet. Rev. 12:261330.
 Kotz, S., T.J. Kozubowski, and K. Podgorski. 2001. The Laplace distribution and generalizations: A revisit with applications to communications, economics, engineering, and finance. Boston: Birkhauser. 349 p.
 Korolev, V. Yu., and A. I. Zeifman. 2016. Anote onmixture representations for the Linnik and MittagLeffler distri
butions and their applications. J. Math. Sci. 218(3):314 327.
 Korolev, V.Yu., and A.I. Zeifman. 2016. Convergence of random sums and statistics constructed from samples with random sizes to the Linnik and MittagLeffler distributions and their generalizations. J. Korean Stat. Soc. Available at: arXiv:1602.02480v1 (accessed December 10, 2016).
 Zolotarev, V. M. 1986. Onedimensional stable distributions. Providence: AMS. 284 p.
 Schneider, W. R. 1986. Stable distributions: Fox function representation and generalization. Stochastic processes in classical and quantum systems. Eds. S. Albeverio, G. Casati, and D. Merlini. Berlin: Springer. 497511.
 Uchaikin, V. V., and V. M. Zolotarev. 1999. Chance and stability. Utrecht: VSP. 570 p.
 Korolev, V. Yu. 2016. Product representations for random variables with the Weibull distributions and their applications. J. Math. Sci. 218(3):298313.
 Tucker, H. 1975. On moments of distribution functions attracted to stable laws. Houston J. Math. 1(1):149152.
 Klebanov, L. B., G. M. Maniya, and I. A. Melamed. 1985. A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables. Theor Probab. Appl. 29(4):791794.
 Klebanov, L. B., andS. T. Rachev. 1996. Sums of a random number of random variables and their approximations with eaccompanying infinitely divisible laws. Serdica 22:471498.
 Bunge, J. 1996. Compositions semigroups and random stability. Ann. Probab. 24:14761489.
 Rachev, S. T. 1991. Probability metrics and the stability of stochastic models. Chichester  New York: Wiley. 494 p.
 Gnedenko, B. V., and V. Yu. Korolev. 1886. Random summation: Limit theorems and applications. Boca Raton: CRC Press. 267 p.
 Korolev, V. Yu. 1995. Convergence of random sequences with the independent random indices I. Theor. Probab. Appl. 39(2):282297.
 Korolev, V. Yu. 1996. Convergence of random sequences with the independent random indices II. Theor. Probab. Appl. 40(4):770772.
 Bening, V. E., and V. Yu. Korolev. 2002. Generalized Pois son models and their applications in insurance and finance. Utrecht: VSP. 434 p.
 Pillai, R. N. 1985. SemiaLaplace distributions. Com mun. Stat. Theor. Meth. 14:9911000.
 Linnik, Yu. V. 1953. Lineynye formy i statisticheskiye kri terii. I, II [Linear forms and statistical criteria. I, II]. Ukr. Math. J. 5(2):207243; 3:247290.
 Laha, R. G. 1961. On a class of unimodal distributions. Proc. Am. Math. Soc. 12:181184.
 Lukacs, E. 1970. Characteristic functions. 2nded. London: Griffin. 350 p.
 Kotz, S., I.V. Ostrovskii, and A. Hayfavi. 1995. Analytic and asymptotic properties of Linnik's probability densities, I. J. Math. Anal. Appl. 193:353371.
 Kotz, S., I.V. Ostrovskii, and A. Hayfavi. 1995. Analytic and asymptotic properties of Linnik's probability densities, II. J. Math. Anal. Appl. 193:497521.
 Sabu, G., and R. N. Pillai. 1987. Multivariate aLaplace distributions. J. Nat. Acad. Math. 5:1318.
 Lin, G. D. 1994. Characterizations of the Laplace and related distributions via geometric compound. Sankhya, A156:19.
 Anderson, D. N. 1992. Amultivariate Linnik distribution. Stat. Probabil. Lett. 14:333336.
 Devroye, L. 1990. A note on Linnik's distribution. Stat. Probabil. Lett. 9:305306.
 Jacques, C., B. Reemillard, and R. Theodorescu. 1999. Estimation of Linnik law parameters. Stat. Decision 17(3):213236.
 Kotz, S., and I.V. Ostrovskii. 1996. A mixture representation of the Linnik distribution. Stat. Probabil. Lett. 26:6164.
 Pakes, A. G. 1998. Mixture representations for symmetric generalized Linnik laws. Stat. Probabil. Lett. 37:213221.
 Gorenflo, R., A.A. Kilbas, F. Mainardi, and S.V. Ro gosin. 2014. MittagLeffler functions, related topics and applications. Berlin  New York: Springer. 420 p.
 Kovalenko, I. N. 1965. On the class of limit distributions for rarefied flows of homogeneous events. Lith. Math. J. 5(4):569573.
 Gnedenko, B. V., and I. N. Kovalenko. 1968. Introduction to queueing theory. Jerusalem: Israel Program for Scientific Translations. 281 p.
 Gnedenko, B. V., and I. N. Kovalenko. 1989. Introduction to queueing theory. 2nd ed. Boston: Birkhauser. 314 p.
 Pillai, R. N. 1990. Harmonic mixtures and geometric infinite divisibility J. Indian Stat. Ass. 28:8798.
 Pillai, R. N. 1990. On MittagLeffler functions and related distributions. Ann. Inst. Stat. Math. 42:157161.
 Weron, K., and M. Kotulski. 1996. On the ColeCole relaxation function and related MittagLeffler distributions. PhysicaA 232:180188.
 Gorenflo, R., and F Mainardi. 2008. Continuous time random walk, MittagLeffler waiting time and fractional diffusion: Mathematical aspects. Anomalous transport: Foundations and applications. Eds. R. Klages, G. Radons, and I. M. Sokolov. Weinheim, Germany: WileyVCH. 93127.
 Gnedenko, B. V., and H. Fahim. 1969. Ob odnoy teoreme perenosa [About one transfer theorem]. Dokl. USSRAkad. Nauk 187(1):1517.
 Reenyi, A. 1956. A Poissonfolyamat egy jellemzese. Maguar Tud. Acad. Mat. Int. Kozl. 1:519527.
 Lim, S. C., and L. P. Teo. 2010. Analytic and asymptotic properties of multivariate generalized Linnik's probability densities. J. Fourier Anal. Appl. 16:715747.
 Korolev, V.Yu., L. Kurmangazieva, and A.I. Zeifman.
2016. On asymmetric generalization of the Weibull distribution by scalelocation mixing of normal laws. J. Korean Stat. Soc. 45:238249. arXiv:1506.06232.
 BarndorffNielsen, O. E. 1977. Exponentially decreasing distributions for the logarithm of particle size. Proc. Roy. Soc. Lond. A 353:401419.
 BarndorffNielsen, O. E., J. Kent, andM. S0rensen. 1982. Normal variancemean mixtures and zdistributions. Int. Stat. Rev. 50(2):145159.
 BarndorffNielsen, O. E. 1978. Hyperbolic distributions and distributions ofhyperbolae. Scand. J. Stat. 5:151157.
 Korolev, V. Yu., and I. A. Sokolov. 2012. Skoshen nye raspredeleniya St'yudenta, dispersionnye gamma raspredeleniya i ikh obobshcheniya kak asimptoticheskie approksimatsii [Skewed Student's distributions, variance gamma distributions, and their generalizations as asymptotic approximations]. Informatika i ee Primeneniya  Inform. Appl. 6(1):210.
 Zaks, L. M., and V. Yu. Korolev. 2013. Obobshchennye dispersionnye gammaraspredeleniya kak predel'nye dlya sluchaynykh summ [Generalized variance gamma distributions as limiting for random sums]. Informatika i ee Primeneniya  Inform. Appl. 7(1):105115.
 Korolev, V. Yu. 2014. Generalized hyperbolic laws as limit distributions for random sums. Theor. Probab. Appl. 58(1):6375.
 Erdogan, M. B., and I.V. Ostrovskii. 1998. Analytic and asymptotic properties of generalized Linnik probability densities. J. Math. Anal. Appl. 217:555578.
 Erdogan, M.B., and I.V. Ostrovskii. 1998. On mixture representation of the Linnik density. J. Aust. Math. Soc. A 64:317326.
 Kalashnikov, V. V. 1997. Geometric sums: Bounds for rare events with applications. Dordrecht: Kluwer Academic Publs. 270 p.
 Pakes, A. G. 1992. A characterization of gamma mixtures of stable laws motivated by limit theorems. Stat. Neerl. 23:209218.
 Kozubowski, T. J. 1998. Mixture representation of Linnik distribution revisited. Stat. Probabil. Lett. 38:157160.
 Kozubowski, T. J. 1999. Exponential mixture representation of geometric stable distributions. Ann. Inst. Stat. Math. 52(2):231238.
[+] About this article
Title
ASYMMETRIC LINNIK DISTRIBUTIONS AS LIMIT LAWS FOR RANDOM SUMS OF INDEPENDENT RANDOM VARIABLES WITH FINITE VARIANCES
Journal
Informatics and Applications
2016, Volume 10, Issue 4, pp 2133
Cover Date
20161230
DOI
10.14357/19922264160403
Print ISSN
19922264
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
Linnik distribution; Laplace distribution; MittagLeffler distribution; normal distribution; scale mixture; normal variancemean mixture; stable distribution; geometrically stable distribution
Authors
V. Yu. Korolev , ,
A. I. Zeifman , , , ,
and A. Yu. Korchagin
Author Affiliations
Faculty of Computational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University, 152 Leninskiye Gory, GSP1, Moscow 119991, Russian Federation
Institute of Informatics Problems, Federal Research Center “Computer Sciences and Control” of the Russian
Academy of Sciences, 442 Vavilov Str.,Moscow 119333, Russian Federation
Vologda State University, 15 Lenin Str., Vologda 160000, Russian Federation
ISEDT RAS, 56A Gorky Str., Vologda 16001, Russian Federation
