Negative Association - Definition, Properties, and Applications

In these notes we present the notion of Negative Association, discuss some of its useful properties, and end with some example applications. The slogan to bear in mind here is “independent, or better”. 1 Negative Association Definition In randomized algorithms, our randomness often takes on the form of independent random variables, allowing us to apply powerful theorems concerning such variables, a prominent example being Chernoff-Hoeffding bounds. However, we can’t always expect random variables we observe (or generate during the run of our algorithms) to be independent. Nonetheless, these variables may satisfy some form of negative dependence, in which case useful properties of independence may carry over. This talk focuses on one such notion of negative dependence, namely Negative Association. Intuition Consider a set of random variables X1, X2, . . . , Xn, satisfying the following: if a subset S of these variables is “high”, then a disjoint subset T must be “low”. This property can be formalized as follows. Definition 1 (Negative Association [10, 8]). A set of random variables X1, X2, . . . , Xn is said to be negatively associated (NA) if for any two disjoint index sets I, J ⊆ [n] and two functions f, g both monotone increasing or both monotone decreasing, it holds E[f(Xi : i ∈ I) · g(Xj : j ∈ J)] ≤ E[f(Xi : i ∈ I)] · E[g(Xj : j ∈ J)]. In order to simplify notation later, we will think of monotone functions f and g and disjoint subsets I, J ⊆ [n] as defining functions in n variables fI , gJ : Rn → R, applying them to vectors ~ X = (X1, X2, . . . , Xn) given by sets of (NA) random variables, and stipulate that the values of f( ~ X) and g( ~ X) be determined by disjoint subsets of the Xi. In this notation, the above definition can be restated as follows. Definition 2 (Negative Association [10, 8]). A set of random variables X1, X2, . . . , Xn is said to be negatively associated (NA) if for any two n-dimensional functions f, g : Rn → R, depending on disjoint subsets of indices and both monotone increasing or both monotone decreasing in their respective indices, it holds E[f( ~ X) · g( ~ X)] ≤ E[f( ~ X)] · E[g( ~ X)]. 2 Useful Properties Part 1 As a special case of the definition of NA, taking fi( ~ X) = Xi, we find that NA variables are negatively correlated. Corollary 1 (NA implies Negative Correlation). Let X1, X2, . . . , Xn be NA random variables. Then, for all i 6= j, the following holds: E[XiXj ] ≤ E[Xi] · E[Xj ]. That is, Cov(Xi, Xj) ≤ 0. Another useful property of NA variables is Negative Orthant Dependence (NOD), given below.