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P. Massart 317Hence by Talagrand’s convex distance inequality (28.4), one gets)P (f ≥ M + z) ≤ 2exp(− z24v(28.6)whenever M is a median of f under P . The preceding inequality applies to aRademacher process, which is a special case of an empirical process. Indeed,setting X = {−1, 1} n and definingf (x) =supt∈Tn∑α i,t x i =i=1n∑i=1α i,t ∗ (x)x iin terms of real numbers α i,t ,onecanseethat,foreveryx and y,n∑n∑ ∣f(x) − f(y) ≤ α i,t ∗ (x)(x i − y i ) ≤ 2∣ 1(x i ≠ y i ).i=1i=1∣α i,t ∗ (x)This means that the function f satisfies the regularity condition (28.5) withα i (x) = 2|α i,t ∗ (x)|. ThusifX = (X 1,t ,...,X n,t )isuniformlydistributedon the hypercube {−1, 1} n ,itfollowsfrom(28.6)thatthesupremumoftheRademacher processZ =supt∈Tn∑α i,t X i,t = f(X)i=1satisfies the sub-Gaussian tail inequalityPr(Z ≥ M + z) ≤ 2exp) (− z2,4vwhere the variance factor v can be taken as v =4sup t∈T (α 2 1,t + ···+ α 2 n,t).This illustrates the power of Talagrand’s convex distance inequality. Alas,while condition (28.5) is perfectly suited for the analysis of Rademacher processes,it does not carry over to more general empirical processes.At first, I found it a bit frustrating that there was no analogue of theGaussian concentration inequality for more general empirical processes andthat Talagrand’s beautiful results were seemingly of no use for dealing withsuprema of empirical processes like (28.2). Upon reading Talagrand (1994)carefully, however, I realized that it contained at least one encouraging result.Namely, Talagrand (1994) proved a sub-Gaussian Bernstein type inequalityfor Z − C E(Z), where C is a universal constant. Of course in Talagrand’sversion, C is not necessarily equal to 1 but it was reasonable to expect thatthis should be the case. This is exactly what Lucien Birgé and I were ableto show. We presented our result at the 1994 workshop organized at Yale inhonor of Lucien Le Cam. A year later or so, I was pleased to hear from MichelTalagrand that, motivated in part by the statistical issues described above,and at the price of some substantial deepening of his approach to concentrationof product measures, he could solve the problem and obtain his now famousconcentration inequality for the supremum of a bounded empirical process;see Talagrand (1996).