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On a Class of Distributions Stable Under Random Summation

  • L. B. Klebanov (a1), A. V. Kakosyan (a2), S. T. Rachev (a3) and G. Temnov (a4)

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.

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Copyright

Corresponding author

Postal address: Department of Probability and Statistics, Charles University, Prague Sokolovska 83, Prague-8, CZ 18675, Czech Republic. Email address: klebanov@chello.cz
∗∗ Postal address: School of Mathematical Sciences, University College Cork, Western Gateway Building, Western Road, Cork, Ireland. Email address: g.temnov@ucc.ie

References

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Keywords

MSC classification

On a Class of Distributions Stable Under Random Summation

  • L. B. Klebanov (a1), A. V. Kakosyan (a2), S. T. Rachev (a3) and G. Temnov (a4)

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