2DkcTXceO

316 Anon-asymptoticwalkto Cirel’son et al. (1976). At any rate, it seemed to me to be somewhat ofa beautiful but isolated mountain, given the abundance of results by MichelTalagrand on the concentration of product measures. In the context of empiricalprocesses, the Gaussian Concentration Theorem implies the followingspectacular result.Assume that X 1,t ,...,X n,t are Gaussian random vectors centered at theirexpectation. Let v be the maximal variance of X 1,t + ···+ X n,t when t varies,and use M to denote either the median or the mean of Z. ThenPr (Z ≥ M + z) ≤ exp(− z22vIn the non-Gaussian case, the problem becomes much more complex. Oneof Talagrand’s major achievements on the topic of concentration inequalitiesfor functions on a product space X = X 1 ×···×X n is his celebrated convexdistance inequality. Given any vector α =(α 1 ,...,α n ) of non-negativereal numbers and any (x, y) ∈X×X,theweightedHammingdistanced α isdefined byn∑d α (x, y) = α i 1(x i ≠ y i ),i=1where 1(A) denotes the indicator of the set A. Talagrand’sconvexdistancefrom a point x to some measurable subset A of X is then defined byd T (x, A) = sup|α| 2 2 ≤1 d α (x, A),where |α| 2 2 = α1 2 + ···+ αn.2If P denotes some product probability measure P = µ 1 ⊗···⊗µ n on X ,the concentration of P with respect to d T is specified by Talagrand’s convexdistance inequality, which ensures that for any measurable set A, one hasP {d T (·,A) ≥ z} ≤P (A)exp).(− z24). (28.4)Typically, it allows the analysis of functions that satisfy the regularity conditionn∑f(x) − f(y) ≤ α i (x)1(x i ≠ y i ). (28.5)i=1One can then play the following simple but subtle game. Choose A = {f ≤M} and observe that in view of condition (28.5), one has f(x)