Variance-generalized-gamma-distributions as limit laws for random sums

A general 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 variance-mean mixtures of normal laws. As a corollary, necessary and sufficient conditions for the convergence of the distributions of sums of a random number of independent identically distributed random variables to variance-generalized-gamma-distributions are obtained. For a special case of continuous-time random walks generated by compound doubly stochastic Poisson processes, convergence rate estimates are presented.