An “FKG equality” with applications to random environments

Abstract

We consider a sequence of independent random variables, X 1, X 2, X 3,…, taking values in {1,2,…, m}. We introduce a σ-algebra of “nonpivotal” events and prove the following 0–1 Law: P( A)=0 or 1 if and only if A is nonpivotal. All tail events are nonpivotal. The proof is based on an “FKG equality” which provides exact error terms to the FKG inequality. We give some applications for independent random variables in a random environment in the sense that the probabilities of particular outcomes are random.