Characterization of discrete laws via mixed sums and Markov branching processes

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Abstract

Let (Z(t)) be a subordinator independent of 0 less than or equal to U less than or equal to 1 and let u and v be positive constants. Solutions to the ''in law'' equation Z(u) = (d)UZ(u+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.
Original languageEnglish
Pages (from-to)285-300
JournalStochastic Processes and their Applications
Volume55
DOIs
Publication statusPublished - 1995

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