## Electronic Communications in Probability

### Transportation Approach to Some Concentration Inequalities in Product Spaces

#### Abstract

Using a transportation approach we prove that for every probability measures $P,Q_1,Q_2$ on $\Omega^N$ with $P$ a product measure there exist r.c.p.d. $\nu_j$ such that $\int \nu_j (\cdot|x) dP(x) = Q_j(\cdot)$ and $$\int dP (x) \int \frac{dP}{dQ_1} (y)^\beta \frac{dP}{dQ_2} (z)^\beta (1+\beta (1-2\beta))^{f_N(x,y,z)} d\nu_1 (y|x) d\nu_2 (z|x) \le 1 \;,$$ for every $\beta \in (0,1/2)$. Here $f_N$ counts the number of coordinates $k$ for which $x_k \neq y_k$ and $x_k \neq z_k$. In case $Q_1=Q_2$ one may take $\nu_1=\nu_2$. In the special case of $Q_j(\cdot)=P(\cdot|A)$ we recover some of Talagrand's sharper concentration inequalities in product spaces.

#### Article information

Source
Electron. Commun. Probab., Volume 1 (1996), paper no. 9, 83-90.

Dates
Accepted: 24 October 1996
First available in Project Euclid: 25 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1453756500

Digital Object Identifier
doi:10.1214/ECP.v1-979

Mathematical Reviews number (MathSciNet)
MR1423908

Zentralblatt MATH identifier
0916.28003

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 28A35: Measures and integrals in product spaces

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