Uniform Common Randomness Generation Over Arbitrary Point-to-Point Channels

Abstract

We consider a standard two-source model for uniform common randomness (UCR) generation, in which two terminals (Terminal A and Terminal B) observe independent and identically distributed (i.i.d.) samples of a correlated finite source, and where Terminal A is allowed to send information to Terminal B over an arbitrary single-user channel. We provide a general theoretical framework, from which the proofs of a general formula for the UCR capacity and general bounds on the $\epsilon $ -UCR capacity of the specified model follow as special cases. The UCR capacity is defined as the maximum achievable CR rate such that the two terminals can agree on a common uniform or nearly uniform random variable with probability arbitrarily close to one, whereas the $\epsilon $ -UCR capacity is defined as the maximum CR rate that can be achieved such that the probability that the two terminals do not agree on a common uniform or nearly uniform random variable does not exceed $\epsilon $ , where $0 \lt \epsilon \lt 1$ is fixed. The established general formula for the UCR capacity depends on a formula characterizing the transmission capacity of arbitrary point-to-point channels, as elaborated by Verdú and Han. The derived general lower and upper bounds on the $\epsilon $ -UCR capacity depend on corresponding lower and upper bounds on the $\epsilon $ -transmission capacity, as proved by Verdú and Han for arbitrary point-to-point channels. Since we are considering general channels, the derived bounds are equal except possibly for at most countably many points, where discontinuity issues might arise. We further provide two examples of channels for which the established bounds are equal and investigate the left-continuity and monotonicity of the derived lower bound.