Concentration inequalities for suprema of unbounded empirical processes
[Inégalités de concentration pour les suprema de processus empiriques non bornés]
Marchina, Antoine1
1 Université de Paris, CNRS, MAP5 UMR 8145, F-75006 Paris, (France)
Annales Henri Lebesgue, Tome 4 (2021), pp. 831-861.

Dans cet article, nous donnons des inégalités de concentration pour des suprema de processus empiriques non bornés et (éventuellement) non centrés, associés à des variables aléatoires indépendantes et identiquement distribuées. En particulier, nous établissons des inégalités de type Fuk–Nagaev avec constantes optimales dans la bande des moyennes déviations. Notre approche est basée sur des techniques de martingales et des inégalités de comparaison permettant de majorer des quantiles généralisés comme la CVaR. Nous étendons également les inégalités de concentration à gauche de Klein (2002) à des classes de fonctions non bornées.

In this paper, we provide new concentration inequalities for suprema of (possibly) non-centered and unbounded empirical processes associated with independent and identically distributed random variables. In particular, we establish Fuk–Nagaev type inequalities with the optimal constant in the moderate deviation bandwidth. The proof builds on martingale methods and comparison inequalities, allowing to bound generalized quantiles as the so-called Conditional Value-at-Risk. Importantly, we also extent the left concentration inequalities of Klein (2002) to classes of unbounded functions.

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Accepté le :
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DOI : 10.5802/ahl.90
Classification : 60E15
Mots clés : concentration inequalities, empirical processes, martingale method, generalized moments
Marchina, Antoine 1

1 Université de Paris, CNRS, MAP5 UMR 8145, F-75006 Paris, (France) 
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Marchina, Antoine. Concentration inequalities for suprema of unbounded empirical processes. Annales Henri Lebesgue, Tome 4 (2021), pp. 831-861. doi : 10.5802/ahl.90. http://www.numdam.org/articles/10.5802/ahl.90/

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