A note on Talagrand's variance bound in terms of influences

Let X_1,..., X_n be independent Bernoulli random variables and $f$ a function on {0,1}^n. In the well-known paper (Talagrand1994) Talagrand gave an upper bound for the variance of f in terms of the individual influences of the X_i's. This bound turned out to be very useful, for instance in percolation theory and related fields. In many situations a similar bound was needed for random variables taking more than two values. Generalizations of this type have indeed been obtained in the literature (see e.g. (Cordero-Erausquin2011), but the proofs are quite different from that in (Talagrand1994). This might raise the impression that Talagrand's original method is not sufficiently robust to obtain such generalizations. However, our paper gives an almost self-contained proof of the above mentioned generalization, by modifying step-by-step Talagrand's original proof.Comment: 10 page

Geometric influences

We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove analogs of the Kahn-Kalai-Linial (KKL) and Talagrand's influence sum bounds for the new definition. We further prove an analog of a result of Friedgut showing that sets with small "influence sum" are essentially determined by a small number of coordinates. In particular, we establish the following tight analog of the KKL bound: for any set in $\mathbb{R}^n$ of Gaussian measure $t$, there exists a coordinate $i$ such that the $i$th geometric influence of the set is at least $ct(1-t)\sqrt{\log n}/n$, where $c$ is a universal constant. This result is then used to obtain an isoperimetric inequality for the Gaussian measure on $\mathbb{R}^n$ and the class of sets invariant under transitive permutation group of the coordinates.Comment: Published in at http://dx.doi.org/10.1214/11-AOP643 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org