Communication and distributional complexity of joint probability mass functions

The problem of truly-lossless (P_e=0) distributed source coding requires knowledge of the joint statistics of the sources. In particular the locations of the zeroes of the probability mass functions (pmfs) are crucial for encoding at rates below (H(X),H(Y)). We consider the distributed computation of the empirical joint pmf P_n of a sequence of random variable pairs observed at physically separated nodes of a network. We consider both worst-case and average measures of information exchange and treat both exact calculation of P_n and a notion of approximation. We find that in all cases the communication cost grows linearly with the size of the input. Further, we consider the problem of determining whether the empirical pmf has a zero in a particular location and show that in most cases considered this also requires a communication cost that is linear in the input size