@booklet {larrygoldstein-fbiatdwa2004,
	title = {Functional BRK Inequalities, and Their Duals, with Applications},
	howpublished = {Discussion Papers},
	year = {2004},
	month = {11},
	publisher = {Journal of Theoretical Probability 20, 275-293 (2007)},
	abstract = {The inequality conjectured by van den Berg and Kesten in [9], and proved by Reimerin [6], states that for A and B events on S, a product of finitely many finite sets, and P any product measure on S, P(A{\textquestiondown}B) P(A)P(B), where A{\textquestiondown}B are the elementary events which lie in both A and B for {\textquoteleft}disjoint reasons.{\textquoteright} This inequality on events is the special case, for indicator functions, of the inequalityhaving the following formulation. Let X be a random vector with n independent components, each in some space Si (such as{\quotedblbase}d), and set S = {\textasciicircum}ni=1Si. Say that the function f : S {\textdagger}{\textquoteright}{\quotedblbase}depends on K {\v S}{\quotesinglbase} 1,...,n if f(x) = f(y) whenever xi = yi for all i {\textasciicircum}{\textasciicircum} K. Then for any given finite or countable collections of non-negative real valued functions f$\pm$$\pm${\textasciicircum}{\textasciicircum}A, g{\texttwosuperior}{\texttwosuperior}{\textasciicircum}{\textasciicircum}B on S which depend on K$\pm$ and L{\texttwosuperior} respectively,EsupK$\pm${\textasciicircum}\copyrightL{\texttwosuperior}={\textasciicircum}{\l}dots} f$\pm$(X) g{\texttwosuperior}(X) Esup f$\pm$(X) Esup g{\texttwosuperior}(X). Related formulations, and functional versions of the dual inequality on events by Kahn,Saks, and Smyth [4], are also considered. Applications include order statistics, assignment problems, and paths in random graphs.},
	url = {/files/ Revision of September 12 2015.pdf, dp374.pdf},
	author = {Larry Goldstein, Yosef Rinott}
}
