Optimal $L_{}^1$-bounds for submartingales
HTML articles powered by AMS MathViewer
- by Lutz Mattner and Uwe Rösler PDF
- Proc. Amer. Math. Soc. 139 (2011), 735-746 Request permission
Abstract:
The optimal function $f$ satisfying \[ \mathbb {E} |\sum _{1}^n X_i | \ge f(\mathbb {E}|X_1|,\ldots ,\mathbb {E}|X_n|) \] for every martingale $(X_1,X_1+X_2, \ldots ,\sum _{i=1}^n X_i)$ is shown to be given by \[ f(a) = \max \Big \{ a_k-\sum _{i=1}^{k-1} a_i\Big \}_{k=1}^n \cup \Big \{\frac {a_k}2\Big \} _{k=3}^n \] 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
- David C. Cox and J. H. B. Kemperman, Sharp bounds on the absolute moments of a sum of two i.i.d. random variables, Ann. Probab. 11 (1983), no. 3, 765–771. MR 704563
- J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
- Kemperman, J.H.B. and Smit, J.C. (1974). Sharp upper and lower bounds for the moments of a martingale. Adv. Appl. Prob. 6 244–245.
- Lutz Mattner, Mean absolute deviations of sample means and minimally concentrated binomials, Ann. Probab. 31 (2003), no. 2, 914–925. MR 1964953, DOI 10.1214/aop/1048516540
- Mattner, L. (2010). Sums of norm spheres are norm shells and lower triangle inequalities are sharp. Math. Semesterber. 57 11–16.
- Iosif Pinelis, Spherically symmetric functions with a convex second derivative and applications to extremal probabilistic problems, Math. Inequal. Appl. 5 (2002), no. 1, 7–26. MR 1880267, DOI 10.7153/mia-05-02
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
- Bengt von Bahr and Carl-Gustav Esseen, Inequalities for the $r$th absolute moment of a sum of random variables, $1\leq r\leq 2$, Ann. Math. Statist. 36 (1965), 299–303. MR 170407, DOI 10.1214/aoms/1177700291
Additional Information
- Lutz Mattner
- Affiliation: Fachbereich IV - Mathematik, Universität Trier, 54286 Trier, Germany
- MR Author ID: 315405
- Email: mattner@uni-trier.de
- Uwe Rösler
- Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, 24098 Kiel, Germany
- Email: roesler@math.uni-kiel.de
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 735-746
- MSC (2000): Primary 60G42, 60E15; Secondary 26B25
- DOI: https://doi.org/10.1090/S0002-9939-2010-10548-8
- MathSciNet review: 2736352