Edgeworth expansions for multivariate random sums

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