We propose some explicit values for the constants involved in the exponential concentration inequalities for empirical processes which are due to Talagrand. It has been shown by Ledoux that deviation inequalities for empirical processes could be obtained by iteration of logarithmic Sobolev type inequalities. Our approach follows closely that of Ledoux. The improvements that we get with respect to Ledoux’s work are based on refinements of his entropy inequalities and computations.
"About the constants in Talagrand's concentration inequalities for empirical processes." Ann. Probab. 28 (2) 863 - 884, April 2000. https://doi.org/10.1214/aop/1019160263