2010/08/18 by Lev B. Klebanov, A. V. Kakosyan, Klebanov, L. B. +5 · 1 citation
Economics, Econometrics and Finance · Mathematics · Decision Sciences · #Stochastic processes and financial applications #Functional Equations Stability Results #Probability and Risk Models
paper · pdf · doi:10.48550/arxiv.1008.3150
We investigate a family of distributions having a property of stability-under-addition, provided that the number ν of added-up random variables in the random sum is also a random variable. We call the corresponding property a ν-stability and investigate the situation with the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we show that the characteristic function of such a ν-stable distribution can be represented in terms of Chebyshev polynomials, and for the case of ν-normal distribution, the resulting characteristic function corresponds to the hyperbolic secant distribution. We discuss some specific properties of the class and present particular examples.