The Common Information of N Dependent Random Variables

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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 squences 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 G\'{a}cs and K\"{o}rner's common randomness.