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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the distribution of maxima of martingales

Authors: Lester E. Dubins and David Gilat
Journal: Proc. Amer. Math. Soc. 68 (1978), 337-338
MSC: Primary 60G45
MathSciNet review: 0494473
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Abstract: Partial order the set of distributions on the real line by $ \nu \leqslant \nu '$ if $ \nu (x,\infty ) \leqslant \nu '(x,\infty )$ for all x. Then, for each $ \mu $ with a finite first moment, the family $ M(\mu )$ of all $ \nu $ which are distributions of (essential) suprema of martingales closed on the right by a $ \mu $-distributed random number, has a least upper bound $ {\mu ^ \ast }$, and is, therefore, a tight family. In fact, $ {\mu ^ \ast }$ is $ \bar \mu $, the distribution of the Hardy-Littlewood extremal maximal function associated with $ \mu $. Moreover, $ {\mu ^ \ast }$ is itself an element of $ M(\mu )$. For each $ p > 1$, the classical moment inequality that the $ {L_p}$ norm of $ \bar \mu $ (and of $ {\mu ^ \ast }$) is at most $ p/(p - 1)$ times the $ {L_p}$ norm of $ \mu $ is shown to be sharp.

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Keywords: Hardy-Littlewood maximal function, martingale, $ {L_p}$, $ L \log L$
Article copyright: © Copyright 1978 American Mathematical Society