Transportation Approach to Some Concen- Tration Inequalities in Product Spaces

Using a transportation approach we prove that for every probability measures P,Q1, Q2 on Ω N 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) dP dQ2 (z)(1 + β(1 − 2β))N dν1(y|x)dν2(z|x) ≤ 1 , for every β ∈ (0, 1/2). Here fN counts the number of coordinates k for which xk 6= yk and xk 6= 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. In [Tal95, Tal96a], Talagrand provides a variety of concentration of measure inequalities which apply in every product space Ω (with Ω Polish) equipped with a Borel product (probability) measure P . These inequalities are extremely useful in combinatorial applications such as the longest common/increasing subsequence, in statistical physics applications such as the study of spin glass models, and in areas touching upon functional analysis such as probability in Banach spaces (c.f. [Tal95, Tal96a] and the references therein). The proofs of these inequalities are all based on an induction on N , where in order to prove the concentration of measure result for a generic set A ⊂ Ω one applies the induction hypothesis for the N dimensional sets A(ω) = {(y1, . . . , yN) : (y1, . . . , yN , ω) ∈ A}, ω ∈ Ω fixed, and B = ∪ωA(ω). Marton, in [Mar96a, Mar96b], building upon [Mar86], extends some of Talagrand’s results to the context of contracting Markov chains. In these works concentration inequalities related to the “distance” between a set A and a point x are viewed as consequences of inequalities 1Partially supported by an NSF DMS-9403553 grant and by a US-ISRAEL BSF grant 2Partially supported by a US-ISRAEL BSF grant and by the Technion E.& J.Bishop research fund