Compound Power Series Distribution with Negative Multinomial Summands

The paper considers a multivariate distribution whose coordinates are compounds. The number of the summands is itself also a multivariate compound with one and the same univariate Power series distributed number of summands and negative multinomially distributed summands. In the total claims amount process the summands are independent identically distributed random vectors. We provide the first full characterization of this distribution. We show that considered as a mixture this distribution would be Mixed Negative multinomial distribution having the possibly scale changed power series distributed THE first parameter. We provide an interesting application to risk theory.