Multivariate compounds with equal number of summands

The paper considers multivariate discrete random sums with equal number of summands. Such distributions describe the total claim amount received by a company in a fixed time point. In Queuing theory they characterize cumulative waiting times of customers up to time t. In Theory of branching processes they model the number of heirs at a fixed point in time. Here some general properties and formulae for numerical characteristics of these distributions are derived and some particular cases are considered.