Transportation Approach to Some Concentration Inequalities in Product Spaces

Using a transportation approach we prove that for every probability measures P,Q₁,Q₂ on ΩN with P a product measure there exist r.c.p.d. νⱼ such that ∫ νⱼ (·|x) dP(x) = Qⱼ(·) and ∫ dP (x) ∫ dP/dQ₁ (y)^β dP/dQ₂ (z)^β (1+β (1-2β))fN(x,y,z) dν₁ (y|x) dν₂ (z|x) ≤ 1 \;, for every β ∈ (0,1/2). Here fN counts the number of coordinates k for which xₖ ≠ yₖ and xₖ ≠ zₖ. In case Q₁=Q₂ one may take ν₁=ν₂. In the special case of Qⱼ(·)=P(·|A) we recover some of Talagrand's sharper concentration inequalities in product spaces.