Martingale Marginals Do Not Always Determine Convergence

Abstract

Baéz-Duarte (J. Math. Anal. Appl. 36, 149–150, 1971, http://dx.doi.org/10.1016/0022-247X(71)90025-4 [ISSN 0022-247x]) and Gilat (Ann. Math. Stat. 43, 1374–1379, 1972, http://dx.doi.org/10.1214/aoms/1177692494 [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.

References

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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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