  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 apply
our geometric and geometric-type approximations to risk processes and related
areas, including the approximation of Poisson processes with a random time
horizon 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.
