Edgeworth expansions for multivariate random sums

The sum of a random number independent and identically distributed vectors has distribution which is not analytically tractable, in the general case. problem been addressed by means asymptotic approximations embedding summands stochastically increasing sequence. Another approach relies on fitting flexible tractable parametric, multivariate distributions, as for example finite mixtures. Both approaches are investigated within framework Edgeworth expansions. A formula fourth-order cumulants derived it shown that above mentioned does necessarily lead to valid normal approximations. theoretical empirical results suggest mixtures two distributions with proportional covariance matrices satisfactorily fit data generated from sums where counting variable Poisson skew-normal, respectively.