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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.
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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 

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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