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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the distribution of maxima of martingales
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by Lester E. Dubins and David Gilat PDF
Proc. Amer. Math. Soc. 68 (1978), 337-338 Request permission

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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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 68 (1978), 337-338
  • MSC: Primary 60G45
  • DOI: https://doi.org/10.1090/S0002-9939-1978-0494473-4
  • MathSciNet review: 0494473