  We introduce two abstract theorems that reduce a variety of complex
exponential distributional approximation problems to the construction of
couplings. These are applied to obtain new rates of convergence with respect to
the Wasserstein and Kolmogorov metrics for the theorem of R\'{e}nyi on random
sums and generalizations of it, hitting times for Markov chains, and to obtain
a new rate for the classical theorem of Yaglom on the exponential asymptotic
behavior of a critical Galton--Watson process conditioned on nonextinction. The
primary tools are an adaptation of Stein's method, Stein couplings, as well as
the equilibrium distributional transformation from renewal theory.
