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On the supremum of certain families of stochastic processes
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Li, Wenbo V., Pillai, Natesh S. 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, Vol.2009 (No.14).
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Official URL: http://www2.warwick.ac.uk/fac/sci/statistics/crism...
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 | ||||
| Official Date: | 2009 | ||||
| Dates: |
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| 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| Status: | Not Peer Reviewed | ||||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) |
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