Properties of Two-Dimensional Maxima of Particle Scores in Critical Branching Processes with Immigration and Continuous Time

We study two-dimensional maxima of particle scores in critical branching processes with immigration and continuous time. The limit distribution for the maxima of two scores at two instants of time is found. We obtain the limit intensities of upward and downward jumps of maxima of one score and the limit intensities of joint upward and downward jumps for both scores or for at least one score. In the case of independent scores, we calculate the average numbers of joint upward and downward jumps of maxima over the whole time period. The results are illustrated with examples.