Abstract
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.
Highlights
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 slightly weaken the definition of the distance m(A, x) introduced in [8], but, essentially, this is what is used in the proof of the concentration inequality for empirical processes
The Theorem 1 below is almost identical to Theorem 4.2 in [8] and we assume that the reader is familiar with the proof
Summary
As we mentioned above the proof is identical to the proof of Theorem 4.2 in [8] where Proposition 4.2 is substituted by the Lemma 1 below, and the case of n = 1 must be adjusted to the new definition of ψα. By the definition of ψα one has to consider two separate cases: If p−1 ≥ 2α The proof of both of these inequalities constitutes a tedious exersize in calculus and is boring enough not to include it in this paper. Note that in order to show the existence of {kji } in the statement of lemma one should try to minimize the left side of (1.4) with respect to {kji } under the constraints (1.3). To make the induction step we will start by proving the following statement, where we assume that kji correspond to the specific optimal choice indicated above.
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