In this paper we revisit Talagrand's proof of concentration inequality for empirical processes. We give a different proof of the main technical lemma that guarantees the existence of a certain kernel. Moreover, we generalize the result of Talagrand to a family of kernels which in one particular case allows us to produce the Poissonian bound without using the truncation argument. We also give some examples of applications of the abstract concentration inequality to empirical processes that demonstrate some interesting properties of Talagrand's kernel method.
"A Note on Talagrand's Concentration Inequality." Electron. Commun. Probab. 6 55 - 65, 2001. https://doi.org/10.1214/ECP.v6-1034