Journal of Applied Probability

On a class of distributions stable under random summation

L. B. Klebanov, A. V. Kakosyan, S. T. Rachev, and G. Temnov

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Abstract

We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.

Article information

Source
J. Appl. Probab. Volume 49, Number 2 (2012), 303-318.

Dates
First available in Project Euclid: 16 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.jap/1339878788

Digital Object Identifier
doi:10.1239/jap/1339878788

Mathematical Reviews number (MathSciNet)
MR2977797

Zentralblatt MATH identifier
1245.60022

Subjects
Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60E10: Characteristic functions; other transforms

Keywords
Stability random summation characteristic function hyperbolic secant distribution

Citation

Klebanov, L. B.; Kakosyan, A. V.; Rachev, S. T.; Temnov, G. On a class of distributions stable under random summation. J. Appl. Probab. 49 (2012), no. 2, 303--318. doi:10.1239/jap/1339878788. https://projecteuclid.org/euclid.jap/1339878788.


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