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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Martingale property of empirical processes


Authors: Sergio Albeverio, Yeneng Sun and Jiang-Lun Wu
Journal: Trans. Amer. Math. Soc. 359 (2007), 517-527
MSC (2000): Primary 60G42, 60G44; Secondary 03H05, 28E05, 60F15
Published electronically: September 19, 2006
MathSciNet review: 2255184
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Abstract: It is shown that for a large collection of independent martingales, the martingale property is preserved on the empirical processes. Under the assumptions of independence and identical finite-dimensional distributions, it is proved that a large collection of stochastic processes are martingales essentially if and only if the empirical processes are also martingales. These two results have implications on the testability of the martingale property in scientific modeling. Extensions to submartingales and supermartingales are given.


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Additional Information

Sergio Albeverio
Affiliation: Institut für Angewandte Mathematik der Universität Bonn, Wegelerstr. 6, D-53115 Bonn, Germany
Email: albeverio@uni-bonn.de

Yeneng Sun
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore
Email: matsuny@nus.edu.sg

Jiang-Lun Wu
Affiliation: Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom
Email: j.l.wu@swansea.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04055-4
PII: S 0002-9947(06)04055-4
Keywords: Essential independence, finite-dimensional distributions, empirical process, exact law of large numbers, Loeb product space, Keisler's Fubini theorem, martingale, submartingale, supermartingale.
Received by editor(s): September 16, 2004
Published electronically: September 19, 2006
Article copyright: © Copyright 2006 American Mathematical Society