A BKR Operation for Events Occurring for Disjoint Reasons with High Probability

Abstract

Given events A and B on a product space S = n i=1 S i , the set A B consists of all vectors x = (x 1 , . . . , x n ) ∈ S for which there exist disjoint coordinate subsets K and L of {1, . . . , n} such that given the coordinates x i , i ∈ K one has that x ∈ A regardless of the values of x on the remaining coordinates, and likewise that x ∈ B given the coordinates x j , j ∈ L. For a finite product of discrete spaces endowed with a product measure, the BKR inequalitywas conjectured by van den Berg and Kesten [3] and proved by Reimer [13]. In [7] inequality (1) was extended to general product probability spaces, replacing A B by the set A 11 B consisting of those outcomes x for which one can only assure with probability one that x ∈ A and x ∈ B based only on the revealed coordinates in K and L as above. A strengthening of the original BKR inequality (1) results, due to the fact that A B ⊆ A 11 B. In particular, it may be the case that A B is empty, while A 11 B is not.We propose the further extension A st B depending on probability thresholds s and t, where A 11 B is the special case where both s and t take the value one. The outcomes x in A st B are those for which disjoint sets of coordinates K and L exist such that given the values of x on the revealed set of coordinates K, the probability that A occurs is at least s, and given the coordinates of x in L, the probability of B is at least t. We provide simple examples that illustrate the utility of these extensions.