We give exponentially small upper bounds on the success probability for computing the direct product of any function over any distribution using a communication protocol. Let suc(mufC) denote the maximum success probability of a 2-party communication protocol for computing f(xy) with C bits of communication, when the inputs (xy) are drawn from the distribution mu. Let mun be the product distribution on n inputs and fn denote the function that computes n copies of f on these inputs.

We prove that if Tlog15TCn and suc(mufC)09 , then suc(munfnT)exp(−(n)) . When mu is a product distribution, we prove a nearly optimal result: as long as Tlog2TCn, we must have suc(munfnT)exp((n)) .