The Essential Equivalence of Pairwise and Mutual Conditional Independence∗

For a large collection of random variables, pairwise conditional independence and mutual conditional independence are shown to be essentially equivalent. Unlike in the finite setting, a large collection of random variables remains essentially conditionally independent under further conditioning. The essential equivalence of pairwise and multiple versions of exchangeability also follows as a corollary. Our proof is based on an iterative extension of Bledsoe and Morse’s completion of a product measure on a pair of measure spaces. ∗Part of this work was done while Peter Hammond was visiting the National University of Singapore in March–April 2004. This version was completed while Yeneng Sun was visiting the University of Illinois at Urbana–Champaign in October 2004 – February 2005. †Department of Economics, Stanford University, Stanford, CA 94305. e-mail: [email protected] ‡Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543; and Department of Economics, National University of Singapore, 1 Arts Link, Singapore 117570. e-mail: [email protected]