Asymptotic Coupling and Its Applications in Information Theory

A coupling of two distributions <inline-formula> <tex-math notation="LaTeX">$P_{X}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$P_{Y}$ </tex-math></inline-formula> is a joint distribution <inline-formula> <tex-math notation="LaTeX">$P_{XY}$ </tex-math></inline-formula> with marginal distributions equal to <inline-formula> <tex-math notation="LaTeX">$P_{X}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$P_{Y}$ </tex-math></inline-formula>. Given marginals <inline-formula> <tex-math notation="LaTeX">$P_{X}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$P_{Y}$ </tex-math></inline-formula> and a real-valued function <inline-formula> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> of the joint distribution <inline-formula> <tex-math notation="LaTeX">$P_{XY}$ </tex-math></inline-formula>, what is its minimum over all couplings <inline-formula> <tex-math notation="LaTeX">$P_{XY}$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$P_{X}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$P_{Y}$ </tex-math></inline-formula>? We study the asymptotics of such coupling problems with different <inline-formula> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula>’s and with <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula> replaced by <inline-formula> <tex-math notation="LaTeX">$X^{n}=(X_{1},\ldots ,X_{n})$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$Y^{n}=(Y_{1},\ldots ,Y_{n})$ </tex-math></inline-formula> where <inline-formula> <tex-math notation="LaTeX">$X_{i}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$Y_{i}$ </tex-math></inline-formula> are i.i. d. copies of random variables <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula> with distributions <inline-formula> <tex-math notation="LaTeX">$P_{X}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$P_{Y}$ </tex-math></inline-formula>, 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 <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> 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 <italic>general</italic> sources and a generalization of this problem, named the <italic>maximal guessing coupling through a channel problem</italic>. 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.