## The Annals of Probability

### On Russo's Approximate Zero-One Law

Michel Talagrand

#### Abstract

Consider the product measure $\mu_p$ on $\{0, 1\}^n$, when 0 $(\operatorname{resp}. 1)$ is given weight $1 - p (\operatorname{resp}. p)$. Consider a monotone subset $A$ of $\{0, 1\}^n$. We give a precise quantitative form to the following statement: if $A$ does not depend much on any given coordinate, $d\mu_p(A)/dp$ is large. Thus, in that case, there is a threshold effect and $\mu_p(A)$ jumps from near 0 to near 1 in a small interval.

#### Article information

Source
Ann. Probab., Volume 22, Number 3 (1994), 1576-1587.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176988612

Digital Object Identifier
doi:10.1214/aop/1176988612

Mathematical Reviews number (MathSciNet)
MR1303654

Zentralblatt MATH identifier
0819.28002

JSTOR