Functional van den Berg-Kesten-Reimer Inequalities and their Duals, with Applications

TL;DR

This paper extends the BKR inequality to a functional form on finite product spaces, introduces dual inequalities, and explores applications in order statistics, assignment problems, and random graph paths.

Contribution

It provides a novel functional extension of the BKR inequality and its duals, broadening their applicability and connecting them to various probabilistic models.

Findings

Extended BKR inequality to general finite product measure spaces.

Derived dual inequalities related to the original BKR inequality.

Applied the inequalities to problems in order statistics, assignment, and random graphs.

Abstract

The BKR inequality conjectured by van den Berg and Kesten in [11], and proved by Reimer in [8], states that for $A$ and $B$ events on $S$, a finite product of finite sets $S_i,i=1,\ldots,n$, and $P$ any product measure on $S$, $$ P(A \Box B) \le P(A)P(B),$$ where the set $A \Box B$ consists of the elementary events which lie in both $A$ and $B$ for `disjoint reasons.' Precisely, with ${\bf n}:=\{1,\ldots,n\}$ and $K \subset {\bf n}$, for ${\bf x} \in S$ letting $[{\bf x}]_K=\{{\bf y} \in S: y_i = x_i, i \in K\}$, the set $A \Box B$ consists of all ${\bf x} \in S$ for which there exist disjoint subsets $K$ and $L$ of ${\bf n}$ for which $[{\bf x}]_K \subset A$ and $[{\bf x}]_L \subset B$. The BKR inequality is extended to the following functional version on a general finite product measure space $(S,\mathbb{S})$ with product probability measure $P$, $$E\left\{ \max_{\stackrel{K \cap L = \emptyset}{K \subset {\bf n}, L \subset {\bf n}}} \underline{f}_K({\bf X})\underline{g}_L({\bf X})\right\} \leq E\left\{f({\bf X})\right\}\,E\left\{g({\bf X})\right\},$$ where $f$ and $g$ are non-negative measurable functions, $\underline{f}_K({\bf x}) = {\rm ess} \inf_{{\bf y} \in [{\bf x}]_K}f({\bf y})$ and $\underline{g}_L({\bf x}) = {\rm ess} \inf_{{\bf y} \in [{\bf x}]_L}g({\bf y}).$ The original BKR inequality is recovered by taking $f({\bf x})={\bf 1}_A({\bf x})$ and $g({\bf x})={\bf 1}_B({\bf x})$, and applying the fact that in general ${\bf 1}_{A \Box B} \le \max_{K \cap L = \emptyset} \underline{f}_K({\bf x}) \underline{g}_L({\bf x})$. Related formulations, and functional versions of the dual inequality on events by Kahn, Saks, and Smyth [6], are also considered. Applications include order statistics, assignment problems, and paths in random graphs.