Distributional properties of jumps of multi-type CBI processes

We study the distributional properties of jumps of multi-type continuous state and continuous time branching processes with immigration (multi-type CBI processes). We derive an expression for the distribution function of the first jump time of a multi-type CBI process with jump size in a given Borel set having finite total L\'evy measure, which is defined as the sum of the measures appearing in the branching and immigration mechanisms of the multi-type CBI process in question. Using this we derive an expression for the distribution function of the local supremum of the norm of the jumps of a multi-type CBI process. Further, we show that if $A$ is a nondegenerate rectangle anchored at zero and with total L\'evy measure zero, then the probability that the local coordinate-wise supremum of jumps of the multi-type CBI process belongs to $A$ is zero. We also prove that a converse statement holds.