Asymptotic Coupling and Its Applications in Information Theory

A coupling of two distributions P_X and P_Y is a joint distribution P_XY with marginal distributions equal to P_X and P_Y. Given marginals P_X and P_Y and a real-valued function f of the joint distribution P_XY, what is its minimum over all couplings P_XY of P_X and P_Y? We study the asymptotics of such coupling problems with different f's and with X and Y replaced by Xⁿ = (X₁, . . . , X_n) and Yⁿ = (Y1, . . . , Y_n) where X_i and Y_i are i.i.d. copies of random variables X and Y with distributions P_X and P_Y, respectively. These include the maximal coupling, minimum distance coupling, maximal guessing coupling, and minimum entropy coupling problems. We characterize the limiting values of these coupling problems as n tends to infinity. We show that they typically converge at least exponentially fast to their limits. Moreover, for the problems of maximal coupling and minimum excess-distance probability coupling, we also characterize (or bound) the optimal convergence rates (exponents). Furthermore, for the maximal guessing coupling problem, we show that it is equivalent to the distribution approximation problem. Therefore, some existing results for the latter problem can be used to derive the asymptotics of the maximal guessing coupling problem. We also study the asymptotics of the maximal guessing coupling problem for two general sources and a generalization of this problem, named the maximal guessing coupling through a channel problem. We apply the preceding results to several new information-theoretic problems, including exact intrinsic randomness, exact resolvability, channel capacity with input distribution constraint, and perfect stealth and secrecy communication.