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Note on the Relation Between Compound and Composed Poisson Processes

Published online by Cambridge University Press:  29 August 2014

Carl Philipson
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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.

Type
Astin Colloquium 1965 Lucerne Subject one
Information
ASTIN Bulletin: The Journal of the IAA , Volume 4 , Issue 2 , February 1967 , pp. 191 - 192
Copyright
Copyright © International Actuarial Association 1967