Martingale Marginals Do Not Always Determine Convergence

  • Jim Pitman
Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)


Baéz-Duarte (J. Math. Anal. Appl. 36, 149–150, 1971, [ISSN 0022-247x]) and Gilat (Ann. Math. Stat. 43, 1374–1379, 1972, [ISSN 0003-4851]) gave examples of martingales that converge in probability (and hence in distribution) but not almost surely. Here such a martingale is constructed with uniformly bounded increments, and a construction is provided of two martingales with the same marginals, one of which converges almost surely, while the other does not converge in probability.


Random Walk Marginal Distribution Simple Random Walk Martingale Property Walk Path 
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Thanks to David Aldous for drawing my attention to Gilat [3].


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    F. Hirsch, C. Profeta, B. Roynette, M. Yor, Peacocks and Associated Martingales, with Explicit Constructions. Bocconi & Springer Series, vol. 3 (Springer, Milan, 2011). doi:  10.1007/978-88-470-1908-9. [ISBN 978-88-470-1907-2]
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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of CaliforniaBerkeleyUSA

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