Many statistical procedures are derived from probabilistic inequalities and results; such procedures may need more precise bounds as this is proved in the present chapter for the independent case. Basic notations are those from Appendix B.1. Developments may be found in van der Vaart (Asymptotic statistics, Cambridge University Press, Cambridge, 1998) and those related with functional estimation may be found in the monograph (Rosenblatt, Stochastic curve estimation, NSF-CBMS regional conference series in probability and statistics, vol 3, 1991). We begin the chapter with applications of the moment inequalities in Lemma 2.2.1 which are useful for empirical procedures. Then we describe empirical estimators, contrast estimators and non-parametric estimators. The developments do not reflect the relative interest of the topics but are rather considered with respect to possible developments under dependence conditions hereafter.