TY - JOUR

T1 - Characterization of discrete laws via mixed sums and Markov branching processes

AU - Pakes, Anthony

PY - 1995

Y1 - 1995

N2 - 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.

AB - 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.

U2 - 10.1016/0304-4149(94)00049-Y

DO - 10.1016/0304-4149(94)00049-Y

M3 - Article

VL - 55

SP - 285

EP - 300

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

ER -