## Journal of Applied Probability

### On a class of distributions stable under random summation

#### 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

https://projecteuclid.org/euclid.jap/1339878788

Digital Object Identifier
doi:10.1239/jap/1339878788

Mathematical Reviews number (MathSciNet)
MR2977797

Zentralblatt MATH identifier
1245.60022

#### 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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