Note on the Relation Between Compound and Composed Poisson Processes

Carl Philipson

ASTIN Bulletin, 1967, vol. 4, issue 2, 191-192

Abstract: 1. In an article on such processes in ASTIN Bull. (II-3, pp. 450–451) the author has derived a relation between cPp i.n.s. (compound Poisson processes in the narrow sense) and composed Poisson processes, which is quite general. In this note this relation shall be simplified under the rather weak condition that Po(t), Pn(t) being the distribution of the number of changes occurring in the interval (o, t), for every fixed value of t, tends to a positive limit less than unity when t tends to infinity.2. By the theory expounded by Thyrion (Bull, de l'Assoc. Act. Beiges, 1959 and 1963) the characteristic function of the number of events in the interval (o, t) can for each fixed value of t in this case be written where η is a real variable and i the imaginary unity.It is easily seen that (1) is equal to the following expression which has the form of a characteristic function of the number of changes in a composed Poisson process.

Date: 1967
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