Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T03:15:39.864Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximations for compound Poisson and Pólya processes

Published online by Cambridge University Press:  01 July 2016

Paul Embrechts ,
Jens L. Jensen ,
Makoto Maejima  and
J. L. Teugels
Paul Embrechts*
Affiliation:
Imperial College, London
Jens L. Jensen*
Affiliation:
Aarhus University
Makoto Maejima*
Affiliation:
Keio University
J. L. Teugels*
Affiliation:
Katholieke Universiteit Leuven
*
∗Postal address: Department of Mathematics, Imperial College, London SW7 2BZ, UK.
∗∗Postal address: Department of Theoretical Statistics, Institute of Mathematics, Aarhus University, 8000 Aarhus C, Denmark.
∗∗∗Postal address: Department of Mathematics, Keio University, 3–14–1, Hiyoshi, Kohoku-ku, Yokohama 223, Japan.
∗∗∗∗Postal address: Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, B-3030 Leuven (Heverlee), Belgium.
Get access Rights & Permissions [Opens in a new window]

Abstract

Suppose Xi≧0 are i.i.d., i = 1, 2, ···. We derive a saddlepoint approximation for P{∑N(t)k=1Xk> y} as y→∞ and t is fixed, where N(t), t≧0, is either a Poisson or a Pólya process. These results are then compared and contrasted with the well-known Esscher approximation.

Keywords

ASYMPTOTIC EXPANSIONSCOMPOUND PROCESSESESSCHER APPROXIMATIONRUIN THEORYSADDLEPOINT METHOD
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 3 , September 1985 , pp. 623 - 637
Copyright
Copyright © Applied Probability Trust 1985 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research partly supported by a grant from the Nuffield Foundation.

References

Bohman, H. (1963) What is the reason that Esscher’s method of approximation is as good as it is? Skand. Akt. Tidskr. , 87–94.Google Scholar
Esscher, F. (1932) On the probability function in the collective theory of risk. Skand. Akt. Tidskr. , 175–195.Google Scholar
Esscher, F. (1963) On approximate computations when the corresponding characteristic functions are known. Skand. Akt. Tidskr. , 78–86.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications , Vol. 2. Wiley, New York.Google Scholar
Loève, M. (1963) Probability Theory. Van Nostrand, New York.Google Scholar
Widder, D. V. (1941) The Laplace Transform , Princeton University Press, Princeton, NJ.Google Scholar