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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Martingale property of empirical processes
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by Sergio Albeverio, Yeneng Sun and Jiang-Lun Wu PDF
Trans. Amer. Math. Soc. 359 (2007), 517-527 Request permission


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:
  • Yeneng Sun
  • Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore
  • Email:
  • Jiang-Lun Wu
  • Affiliation: Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom
  • Email:
  • Received by editor(s): September 16, 2004
  • Published electronically: September 19, 2006
  • © Copyright 2006 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 517-527
  • MSC (2000): Primary 60G42, 60G44; Secondary 03H05, 28E05, 60F15
  • DOI:
  • MathSciNet review: 2255184