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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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

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

First available in Project Euclid: 16 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Stability random summation characteristic function hyperbolic secant distribution


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.

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