Families of ν-similar birth-death processes and limiting conditional distributions

We study, for a given process , the conditions under which nu-similarity is provided, relations between similarity and nu-similarity of processes, and we analyze (doubly) limiting conditional distributions. We present sufficient conditions for the existence of at least one nu-similar process to a given one. Moreover, recursive formulas for birth and death rates of nu-similar processes are formulated. We apply differential equations and uniqueness of solutions to such equations to derive the main results of the paper. Finally, an example illustrates the applications of our results.