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

TL;DR

This paper provides a nearly self-contained proof extending Talagrand's variance bound, originally for Bernoulli variables, to variables with more than two values, demonstrating the robustness of his method.

Contribution

It offers a step-by-step modification of Talagrand's original proof to generalize the variance bound to multi-valued variables.

Findings

Extended Talagrand's variance bound to non-binary variables

Provided a robust, self-contained proof of the generalization

Showed the original method's adaptability to broader contexts

Abstract

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.