Informatics and Applications
2016, Volume 10, Issue 4, pp 21-33
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.
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[+] 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 21-33
Cover Date
2016-12-30
DOI
10.14357/19922264160403
Print ISSN
1992-2264
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
Linnik distribution; Laplace distribution; Mittag-Leffler distribution; normal distribution; scale mixture; normal variance-mean 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, 1-52 Leninskiye Gory, GSP-1, Moscow 119991, Russian Federation
Institute of Informatics Problems, Federal Research Center “Computer Sciences and Control” of the Russian
Academy of Sciences, 44-2 Vavilov Str.,Moscow 119333, Russian Federation
Vologda State University, 15 Lenin Str., Vologda 160000, Russian Federation
ISEDT RAS, 56-A Gorky Str., Vologda 16001, Russian Federation
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