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.

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In this paper we estimate and analyze the errors associated with the use of the discrete (fast) Fourier transformation for the numerical calculation of convo-lutions. We suggest and compare methods to reduce these errors without loosing the computational efficiency of the calculation scheme. A typical field of application of our findings is the calculation of aggregate loss distributions for, e.g., losses from insurance cases or operational risk losses in the finance industry.

Typically, operational risk losses are reported above a threshold. Fitting data reported above a constant threshold is a well known and studied problem. However, in practice, the losses are scaled for business and other factors before the fitting and thus the threshold is varying across the scaled data sample. A reporting level may also change when a bank changes its reporting policy. We present both the maximum likelihood and Bayesian Markov chain Monte Carlo approaches to fitting the frequency and severity loss distributions using data in the case of a time varying threshold. Estimation of the annual loss distribution accounting for parameter uncertainty is also presented.

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