  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 $X^{n}=(X_{1},\ldots,X_{n})$
and $Y^{n}=(Y_{1},\ldots,Y_{n})$ 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
\emph{general} sources and a generalization of this problem, named the
\emph{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.
