Compensator and exponential inequalities for some suprema of counting processes

Talagrand [1996. New concentration inequalities in product spaces. Invent. Math. 126 (3), 505–563], Ledoux [1996. On Talagrand deviation inequalities for product measures. ESAIM: Probab. Statist. 1, 63–87], Massart [2000a. About the constants in Talagrand's concentration inequalities for empirical processes. Ann. Probab. 2 (28), 863–884], Rio [2002. Une inégalité de Bennett pour les maxima de processus empiriques. Ann. Inst. H. Poincaré Probab. Statist. 38 (6), 1053–1057. En l’honneur de J. Bretagnolle, D. Dacunha-Castelle, I. Ibragimov] and Bousquet [2002. A Bennett concentration inequality and its application to suprema of empirical processes. C. R. Math. Acad. Sci. Paris 334 (6), 495–500] have obtained exponential inequalities for suprema of empirical processes. These inequalities are sharp enough to build adaptive estimation procedures Massart [2000b. Some applications of concentration inequalities. Ann. Fac. Sci. Toulouse Math. (6) 9 (2), 245–303]. The aim of this paper is to produce these kinds of inequalities when the empirical measure is replaced by a counting process. To achieve this goal, we first compute the compensator of a suprema of integrals with respect to the counting measure. We can then apply the classical inequalities which are already available for martingales Van de Geer [1995. Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes. Ann. Statist. 23 (5), 1779–1801].