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On the supremum of certain families of stochastic processes

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Li, Wenbo V., Pillai, Natesh S., 1981- and Wolpert, Robert L. (2009) On the supremum of certain families of stochastic processes. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. (Working papers).

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Abstract

We consider a family of stochastic processes {X-t(epsilon), t is an element of T} on a metric space T, with a parameter epsilon down arrow 0. We study the conditions under which lim(epsilon -> 0)(P)(sup(r is an element of T)vertical bar X-t(epsilon)vertical bar < delta) = 1 In our main result in Section 2, we find conditions in terms of the covering number N(T, d, delta) that ensure (1.1) holds. Although our technique is based on well known chaining methods, our principal result appears to be new. In Section 3 we discuss briefly the optimality of our hypotheses and compare our theorem with the Kolmogorov criterion for continuity of stochastic processes. In Section 4 we present an application of our main theorem to random fields constructed from Levy random measures.

Item Type:	Working or Discussion Paper (Working Paper)
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Stochastic processes
Series Name:	Working papers
Journal or Publication Title:	STATISTICS & PROBABILITY LETTERS
Publisher:	University of Warwick. Centre for Research in Statistical Methodology
Place of Publication:	Coventry
ISSN:	0167-7152
Date:	2009
Volume:	Vol.2009
Number:	No.14
Number of Pages:	8
Page Range:	pp. 916-921
Status:	Not Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
Funder:	National Science Foundation (U.S.) (NSF), University of Warwick. Centre for Research in Statistical Methodology
Grant number:	DMS-0805929 (NSF), DMS-0757549 (NSF), DMS-0635449 (NSF)
References:	Ledoux, Michel. 1996. Isoperimetry and Gaussian analysis. Pages 165–294 of: Lectures on probability theory and statistics (Saint-Flour, 1994). Lecture Notes in Math., vol. 1648. Berlin: Springer. Pillai, Natesh S., & Wolpert, Robert L. 2008. Posterior Consistency of Bayesian Nonparametric Models Using L´evy Random Field Priors. Discussion Paper 2008-08. Duke University Department of Statistical Science. Rajput, Balram S., & Rosi´nski, Jan. 1989. Spectral representations of infinitely divisible processes. Probab. Theory Related Fields, 82(3), 451–487. Reynaud-Bouret, P. 2006. Compensator and exponential inequalities for some suprema of counting processes. Statist. Probab. Lett., 76(14), 1514–1521. Talagrand, Michel. 2005. The generic chaining. Springer Monographs in Mathematics. Berlin: Springer-Verlag. Upper and lower bounds of stochastic processes. Wolpert, Robert L., & Taqqu, Murad S. 2005. Fractional Ornstein-Uhlenbeck L´evy Processes and the Telecom Process: Upstairs and Downstairs. Signal Processing, 85(8), 1523–1545.
URI:	http://wrap.warwick.ac.uk/id/eprint/5743