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Probability theory
Upper bounds for superquantiles of martingales
Emmanuel Rio1 
1 Université de Versailles, Laboratoire de mathématiques, UMR 8100 CNRS, Bâtiment Fermat, 45 Avenue des Etats-Unis, F-78035 Versailles, France.
Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 813-822.

Let (M n ) n be a discrete martingale in L p for p in ]1,2] or p=3. In this note, we give upper bounds on the superquantiles of M n and the quantiles and superquantiles of M n * =max(M 0 ,M 1 ,...,M n ).

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DOI: 10.5802/crmath.207
Classification: 60E15

Emmanuel Rio 1

1 Université de Versailles, Laboratoire de mathématiques, UMR 8100 CNRS, Bâtiment Fermat, 45 Avenue des Etats-Unis, F-78035 Versailles, France.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Emmanuel Rio. Upper bounds for superquantiles of martingales. Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 813-822. doi : 10.5802/crmath.207. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.207/

[1] Bernard Bercu; Bernard Delyon; Emmanuel Rio Concentration inequalities for sums and martingales, SpringerBriefs in Mathematics, Springer, 2015 | Zbl

[2] David H. Blackwell; Lester E. Dubins A converse to the dominated convergence theorem, Ill. J. Math., Volume 7 (1963), pp. 508-514 | MR | Zbl

[3] Jean-René Chazottes; Sébastien Gouëzel Optimal concentration inequalities for dynamical systems, Commun. Math. Phys., Volume 316 (2012) no. 3, pp. 843-889 | DOI | MR | Zbl

[4] Jérôme Dedecker; Florence Merlevède; Emmanuel Rio Rates of convergence in the central limit theorem for martingales in the non stationary setting (2021) (https://hal.archives-ouvertes.fr/hal-03112369v1)

[5] Lester E. Dubins; David Gilat On the distribution of maxima of martingales, Proc. Am. Math. Soc., Volume 68 (1978) no. 3, pp. 337-338 | MR | Zbl

[6] David Gilat; I. Meilijson A simple proof of a theorem of Blackwell & Dubins on the maximum of a uniformly integrable martingale, Séminaire de Probabilitées, Strasbourg / France XXII (Lecture Notes in Mathematics), Volume 1321, Springer, 1988, pp. 214-216 | DOI | Numdam | MR | Zbl

[7] Larry Goldstein Bounds on the constant in the mean central limit theorem, Ann. Probab., Volume 38 (2010) no. 4, pp. 1672-1689 | MR | Zbl

[8] Antoine Marchina Concentration inequalities for suprema of unbounded empirical processes (2021) (to appear in Annales Henri Lebesgue, https://hal.archives-ouvertes.fr/hal-01545101/) | Zbl

[9] Adam Osekowski Sharp martingale and semimartingale inequalities, Monografie Matematyczne. Instytut Matematyczny PAN (New Series), 72, Birkhäuser/Springer, 2012 | MR | Zbl

[10] Iosif Pinelis On normal domination of (super)martingales, Electron. J. Probab., Volume 11 (2006) no. 39, pp. 1049-1070 | MR | Zbl

[11] Iosif Pinelis An Optimal Three-Way Stable and Monotonic Spectrum of Bounds on Quantiles: A Spectrum of Coherent Measures of Financial Risk and Economic Inequality, Risks, Volume 2 (2014) no. 3, pp. 349-392 | DOI

[12] Iosif Pinelis Best possible bounds of the von Bahr–Esseen type, Ann. Funct. Anal., Volume 6 (2015) no. 4, pp. 1-29 | DOI | MR | Zbl

[13] Iosif Pinelis Exact Rosenthal-type bounds, Ann. Probab., Volume 43 (2015) no. 5, pp. 2511-2544 | MR | Zbl

[14] Emmanuel Rio Exponential inequalities for weighted sums of bounded random variables, Electron. Commun. Probab., Volume 20 (2015), 77, 10 pages | MR | Zbl

[15] Emmanuel Rio About Doob’s inequality, entropy and Tchebichef, Electron. Commun. Probab., Volume 23 (2018), 78, 12 pages | MR | Zbl

[16] Pafnutiĭ L. Tchebichef Sur les valeurs limites des intégrales, Liouville J., Volume 19 (1874), pp. 157-160 | Zbl

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