On geometric-type approximations with applications

We explore two aspects of geometric approximation via a coupling approach to Stein's method. Firstly, we refine precision and increase scope for applications by convoluting the approximating geometric distribution with a simple translation selected based on the problem at hand. Secondly, we give applications to several stochastic processes, including the approximation of Poisson processes with random time horizons and Markov chain hitting times. Particular attention is given to geometric approximation of random sums, for which explicit bounds are established. These are applied to give simple approximations, including error bounds, for the infinite-horizon ruin probability in the compound binomial risk process.