New rates for exponential approximation and the theorems of R\'{e}nyi and Yaglom

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 rates of convergence with respect to the Wasserstein and Kolmogorov metrics for the theorem of Rényi 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 non-extinction. The primary tools are an adaptation of Stein’s method, Stein couplings, as well as the equilibrium distributional transformation from renewal theory.