A General Formula for Uniform Common Randomness Capacity

We generalize the uniform common randomness capacity formula, initially established by Ahslwede and Csisz\'ar for a two-source model for common randomness generation from independent and identically distributed (i.i.d.) discrete sources with unidirectional communication over rate-limited discrete noiseless channels to the case when the one-way communication is over arbitrary single-user channels. In our proof, we will make use of the transmission capacity formula established by Verd\'u and Han for arbitrary point-to-point channels.