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Optimal $ L_{}^1$-bounds for submartingales


Authors: Lutz Mattner and Uwe Rösler
Journal: Proc. Amer. Math. Soc. 139 (2011), 735-746
MSC (2000): Primary 60G42, 60E15; Secondary 26B25
Published electronically: August 30, 2010
MathSciNet review: 2736352
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Abstract: The optimal function $ f$ satisfying

$\displaystyle \mathbb{E} \vert\sum_{1}^n X_i \vert \ge f(\mathbb{E}\vert X_1\vert,\ldots,\mathbb{E}\vert X_n\vert) $

for every martingale $ (X_1,X_1+X_2, \ldots,\sum_{i=1}^n X_i)$ is shown to be given by

$\displaystyle f(a) = \max \Big\{ a_k-\sum_{i=1}^{k-1} a_i\Big\}_{k=1}^n \cup \Big\{\frac {a_k}2\Big\} _{k=3}^n $

for $ a\in{[0,\infty[}^n_{}$. A similar result is obtained for submartingales $ (0,X_1, X_1+X_2,\ldots, \sum_{i=1}^n X_i)$.

The optimality proofs use a convex-analytic comparison lemma of independent interest.


References [Enhancements On Off] (What's this?)

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Additional Information

Lutz Mattner
Affiliation: Fachbereich IV - Mathematik, Universität Trier, 54286 Trier, Germany
Email: mattner@uni-trier.de

Uwe Rösler
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, 24098 Kiel, Germany
Email: roesler@math.uni-kiel.de

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10548-8
Keywords: Convexity, extremal problem, martingale, moment inequalities
Received by editor(s): April 16, 2009
Received by editor(s) in revised form: April 16, 2010
Published electronically: August 30, 2010
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.