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On a class of distributions stable under random summation
We investigate a family of distributions having a property of
stability-under-addition, provided that the number of added-up random
variables in the random sum is also a random variable. We call the
corresponding property a \,-stability and investigate the situation with
the semigroup generated by the generating function of is commutative.
Using results from the theory of iterations of analytic functions, we show that
the characteristic function of such a -stable distribution can be
represented in terms of Chebyshev polynomials, and for the case of -normal
distribution, the resulting characteristic function corresponds to the
hyperbolic secant distribution. We discuss some specific properties of the
class and present particular examples.Comment: 12 pages, 1 figur
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