On convergence of the distributions of random sums to skew exponential power laws

An extension of the class of exponential power distributions (also known as generalized Laplace distributions) to the nonsymmetric case is proposed. The class of skew exponential power distributions (skew generalized Laplace distributions) is introduced as a family of special variance-mean normal mixtures. Expressions for the moments of skew exponential power distributions are given. It is demonstrated that skew exponential power distributions can be used as asymptotic approximations. For this purpose, a theorem is proved establishing necessary and sufficient conditions for the convergence of the distributions of sums of a random number of independent identically distributed random variables to skew exponential power distributions. Convergence rate estimates are presented for a special case of random walks generated by compound doubly stochastic Poisson processes.