The common information of N dependent random variables

This paper generalizes Wyner´s definition of common information of a pair of random variables to that of N random variables. We prove coding theorems that show the same operational meanings for the common information of two random variables generalize to that of N random variables. As a byproduct of our proof, we show that the Gray-Wyner source coding network can be generalized to N source sequences with N decoders. We also establish a monotone property of Wyner´s common information which is in contrast to other notions of the common information, specifically Shannon´s mutual information and Gács and Körner´s common randomness. Examples about the computation of Wyner´s common information of N random variables are also given.