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Understanding the Wiener–Hopf factorization for the simple random walk

Published online by Cambridge University Press:  14 July 2016

Joanne Kennedy
Joanne Kennedy*
Affiliation:
University of Cambridge
*
∗ Present address: Department of Statistics, University of Oxford, South Parks Road, Oxford OX1 3TG, UK.
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Abstract

We give a sample path proof of the well-known Wiener–Hopf identity F = G– + G+ – G– ∗G+ which relates the ladder-height distributions G– and G+ of a simple random walk to the step distribution F. Unlike previous approaches this direct proof is both simple and intuitive.

Keywords

LADDER-HEIGHT DISTRIBUTIONSAMPLE PATH METHODS
Type
Short Communications
Information
Journal of Applied Probability , Volume 31 , Issue 2 , June 1994 , pp. 561 - 563
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

Research supported by the SERC.

References

References

[1] Asmussen, S. (1989) Aspects of matrix Wiener-Hopf factorization in applied probability. Math. Scientist 14, 101–116.Google Scholar
[2] Feller, W. (1971) An Introduction to Probability Theory and its Applications. Wiley, New York.Google Scholar
[3] Kennedy, J. (1990) A probabilistic view of some algebraic results in Wiener-Hopf theory for symmetrizable Markov chains. In Stochastics and Quantum Mechanics, ed. Truman, A. and Davies, I. M. World Scientific, Swansea.Google Scholar

Reference added in proof

[4] Le Gall, J. F. (1989) Marches aléatoires, mouvement brownien et processus de branchement. In Seminaire de Probabilités XXIII, ed. Azéma, J. et al., pp. 258–274. Lecture Notes in Mathematics 1372, Springer-Verlag, Berlin.Google Scholar