Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2025-01-01T01:40:56.294Z Has data issue: false hasContentIssue false
  • English
  • Français

A New Proof of the Wiener-Hopf Factorization via Basu's Theorem

Part of: Stochastic processes

Published online by Cambridge University Press:  04 February 2016

Brian Fralix  and
Colin Gallagher
Brian Fralix*
Affiliation:
Clemson University
Colin Gallagher*
Affiliation:
Clemson University
*
∗ Postal address: Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC 29634, USA.
∗ Postal address: Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC 29634, USA.
Rights & Permissions [Opens in a new window]

Abstract

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 illustrate how Basu's theorem can be used to derive the spatial version of the Wiener-Hopf factorization for a specific class of piecewise-deterministic Markov processes. The classical factorization results for both random walks and Lévy processes follow immediately from our result. The approach is particularly elegant when used to establish the factorization for spectrally one-sided Lévy processes.

Keywords

Wiener-Hopf factorizationBasu's theoremWiener's Tauberian theoremLévy process

MSC classification

Primary: 60G50: Sums of independent random variables; random walks 60G51: Processes with independent increments; L'evy processes
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012 , pp. 876 - 882
Copyright
© Applied Probability Trust 

References

Boos, D. D. and Hughes-Oliver, J. M. (1998). Applications of Basu's theorem. Amer. Statistician 52, 218–221.Google Scholar
Casella, G. and Berger, R. L. (2002). Statistical Inference. Duxbury, Pacific Grove, CA.Google Scholar
DasGupta, A. (2007). Extensions to Basu's theorem, factorizations, and infinite divisibility. J. Statist. Planning Infer. 137, 945–952.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York.Google Scholar
Fralix, B. H., van Leeuwaarden, J. S. H. and Boxma, O. J. (2011). A new Wiener-Hopf identity for a general class of reflected processes. Submitted.Google Scholar
Ghosh, M. (2002). Basu's theorem with applications: a personalistic review. Sankhyā A 64, 509–531.Google Scholar
Ghosh, J. K. and Singh, R. (1966). Unbiased estimation of location and scale parameters. Ann. Math. Statist. 37, 1671–1675.Google Scholar
Goldberg, R. R. (1970). Fourier Transforms. Cambridge University Press.Google Scholar
Greenwood, P. and Pitman, J. (1980). Fluctuation identities for Lévy processes and splitting at the maximum. Adv. Appl. Prob. 12, 893–902.Google Scholar
Kella, O. and Whitt, W. (1992). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29, 396–403.Google Scholar
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Kyprianou, A. E. (2010). The Wiener–Hopf decomposition. In Encyclopedia of Quantitative Finance, John Wiley, Chichester.Google Scholar
Kyprianou, A. E. and Palmowski, Z. (2005). A martingale review of some fluctuation theory for spectrally negative Lévy processes. In Séminaire de Probabilités XXXVIII (Lecture Notes Math. 1857), Springer, Berlin, pp. 16–29.CrossRefGoogle Scholar
Mattner, L. (1993). Some incomplete but boundedly complete location families. Ann. Statist. 21, 2158–2162.Google Scholar
Nguyen-Ngoc, L. and Yor, M. (2005). Some martingales associated to reflected Lévy processes. In Séminaire de Probabilités XXXVIII (Lecture Notes Math. 1857), Springer, Berlin, pp. 42–69.Google Scholar