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Let X_1,..., X_n be independent Bernoulli random variables and $f$ a functionon {0,1}^n. In the well-known paper (Talagrand1994) Talagrand gave an upperbound 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 andrelated fields. In many situations a similar bound was needed for randomvariables taking more than two values. Generalizations of this type have indeedbeen obtained in the literature (see e.g. (Cordero-Erausquin2011), but theproofs are quite different from that in (Talagrand1994). This might raise theimpression that Talagrand's original method is not sufficiently robust toobtain such generalizations. However, our paper gives an almost self-containedproof of the above mentioned generalization, by modifying step-by-stepTalagrand's original proof.

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