On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law

The probability generating function (pgf) B(s) of the limiting conditional law (LCL) of a subcritical Markov branching process (Z_t : t ≥ 0) (MBP) has a certain integral representation and it satisfies B(0) = 0 and B^́(0) > 0. The general problem posed here is the inverse one: If a given pgf B satisfies these two conditions, is it related in this way to some MBP? We obtain some necessary conditions for this to be possible and illustrate the issues with simple examples and counterexamples. The particular case of the Borel law is shown to be the LCL of a family of MBPs and that the probabilities P₁(Z_t = j) have simple explicit algebraic expressions. Exact conditions are found under which a shifted negative-binomial law can be a LCL. Finally, implications are explored for the offspring law arising from infinite divisibility of the correponding LCL.