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

Given events A and B on a product space \(S={\prod }_{i = 1}^{n} S_{i}\), the set \(A \Box B\) consists of all vectors x = (x₁,…,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 inequality

$$ P(A \Box B) \le P(A)P(B) $$

(1)

was conjectured by van den Berg and Kesten (J Appl Probab 22:556–569, 1985) and proved by Reimer (Combin Probab Comput 9:27–32, 2000). In Goldstein and Rinott (J Theor Probab 20:275–293, 2007) inequality Eq. 1 was extended to general product probability spaces, replacing \(A \Box B\) by the set Open image in new windowx 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 Eq. 1 results, due to the fact that Open image in new window\(A \Box B\) is empty, while Open image in new windowOpen image in new windows and t, where Open image in new windows and t take the value one. The outcomes Open image in new windowK 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.