Characterization of discrete laws via mixed sums and Markov branching processes

Let (Zt) be a subordinator independent of 0 ≤ U ≤ 1 and let u and v be positive constants. Solutions to the “in law” equation Zu = dUZu+v exist under certain conditions and they have a distribution function which is continuous on the positive reals. A discrete version of this equation is here formulated in which ordinary multiplication is replaced by a lattice-preserving operation whose definition involves a subcritical Markov branching process. It is shown that the existence, uniqueness and representation theory for the continuous problem transfers to the discrete problem. Specific examples are exhibited, and extension to two-sided discrete laws is explored.