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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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Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion
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by Gang Wang PDF
Proc. Amer. Math. Soc. 112 (1991), 579-586 Request permission

Abstract:

Let $B = {({B_t})_{t \geq 0}}$ be a standard Brownian motion. For $c > 0$, $k > 0$, let \[ \begin {gathered} T(c,k) = \inf \{ t \geq 0:{\max _{s \leq t}}{B_s} - c{B_t} \geq k\} , \hfill \\ {T^*}(c,k) = \inf \{ t \geq 0:{\max _{s \leq t}}|{B_s}| - c|{B_t}| \geq k\} . \hfill \\ \end {gathered} .\] We show that for $c > 0$ and $k > 0$, both $T(c,k)$ and ${T^*}(c,k)$ are finite almost everywhere. Moreover, $T(c,k)$ and ${T^*}(c,k) \in {L^{p/2}}$ if and only if $c < p/(p - 1)$ for $p > 1$, and for all $c > 0$ when $p \leq 1$. These results have analogues for simple random walks. As a consequence, if $T$ is any stopping time of ${B_t}$ such that ${({B_{T \wedge t}})_{t \geq 0}}$ is uniformly integrable, then both of the inequalities \[ \begin {array}{*{20}{c}} {||{{\sup }_{s \leq T}}{B_s}|{|_p} \leq \frac {p}{{p - 1}}||{B_T}|{|_p},} \\ {{{\left \| {{{\sup }_{s \leq T}}\left | {{B_s}} \right |} \right \|}_P} \leq \frac {p}{{p - 1}}{{\left \| {{B_T}} \right \|}_p},} \\ \end {array} \] are sharp. This implies that $q = p/(p - 1)$ is not only the best constant for Doob’s maximal inequality for general martingales but also for conditionally symmetric martingales (in particular, for dyadic martingales), and for Brownian motion.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 112 (1991), 579-586
  • MSC: Primary 60G42; Secondary 60J65
  • DOI: https://doi.org/10.1090/S0002-9939-1991-1059638-8
  • MathSciNet review: 1059638