Records and increases of multivariate extremes of random particle scores in supercritical branching processes with max-linear heredity

The paper continues the author's long-term studies on extremes of random particle scores in branching processes. It is assumed that multiplication of particles is described by an immortal supercritical discrete-time branching process, the particle scores are dependent due to general heredity, and this dependence is a function of the degree of their relationship. The case of heavy-tail distributions of scores is considered. The max-linear model for scores formation is used. We evaluate the limit probabilities of the current generation superiority to the previous generation or all previous generations, in terms of maxima of particle scores.