Hostname: page-component-6bf8c574d5-8gtf8 Total loading time: 0 Render date: 2025-02-26T07:55:51.477Z Has data issue: false hasContentIssue false
  • English
  • Français

On a transformation of the weighted compound Poisson process

Published online by Cambridge University Press:  29 August 2014

Hans Bühlmann  and
Roberto Buzzi
Hans Bühlmann
Affiliation:
Zurich
Roberto Buzzi
Affiliation:
Zurich
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We are using the following terminology—essentially following Feller:

a) Compound Poisson Variable

This is a random variable

where X1, X2, … Xn, … independent, identically distributed (X0 = o) and N a Poisson counting variable

hence

(common) distribution function of the Xj with j ≠ 0 or in the language of characteristic functions

b) Weighted Compound Poisson Variable

This is a random variable Z obtained from a class of Compound Poisson Variables by weighting over λ with a weight function S(λ)

hence

or in the language of characteristic functions

Let [Z(t); t ≥ o] be a homogeneous Weighted Compound Poisson Process. The characteristic function at the time epoch t reads then

It is most remarkable that in many instances φt(u) can be represented as a (non weighted) Compound Poisson Variable. Our main result is given as a theorem.

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 6 , Issue 1 , September 1971 , pp. 42 - 46
Copyright
Copyright © International Actuarial Association 1971

References

[1]Feller, W.: An Introduction to Probability Theory and its Applications Volume 1, Wiley, New York/London 1957 (second edition).Google Scholar
[2]Thyrion, P.: Théorie collective du risque, Filip Lundberg Colloquium 1968.Google Scholar
[3]Ammeter, H.: A generalization of the collective theory of risk in regard to fluctuating basic probabilities Skandinavisk Aktuarietdskrift, 1948 (volume 31).CrossRefGoogle Scholar