Functional BKR Inequalities, and their Duals, with Applications

Abstract

The inequality conjectured by van den Berg and Kesten (J. Appl. Probab. 22, 556–569, 1985), and proved by Reimer (Comb. Probab. Comput. 9, 27–32, 2000), states that for A and B events on S, a finite product of finite sets, and P any product measure on S,

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

where the set A□B consists of the elementary events which lie in both A and B for ‘disjoint reasons.’ This inequality on events is the special case, for indicator functions, of the inequality having the following formulation. Let X be a random vector with n independent components, each in some space S _i (such as R ^d), and set S=∏ ⁿ _i=1 S _i . Say that the function f:S→R depends on K⊆{1,…,n} if f(x)=f(y) whenever x _i =y _i for all i∈K. Then for any given finite or countable collections of non-negative real valued functions $\{f_{\alpha}\}_{\alpha \in \mathcal{A}},\,\{g_{\beta}\}_{\beta \in \mathcal{B}}$ on S, depending on $K_{\alpha},\alpha \in \mathcal{A}$ and L _β ,β∈ℬ respectively,

$$E\Bigl\{\sup_{K_{\alpha}\cap L_{\beta}=\emptyset}f_{\alpha}(\mathbf{X})g_{\beta}(\mathbf{X})\Bigr\}\leq E\Bigl\{\sup_{\alpha}f_{\alpha}(\mathbf{X})\Bigr\}E\Bigl\{\sup_{\beta}g_{\beta}(\mathbf{X})\Bigr\}.$$

Related formulations, and functional versions of the dual inequality on events by Kahn, Saks, and Smyth (15th Annual IEEE Conference on Computational Complexity 98–103, IEEE Computer Soc., Los Alamitos, CA, 2000), are also considered. Applications include order statistics, assignment problems, and paths in random graphs.