## Journal of Applied Probability

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

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

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

#### 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**

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