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We explore two aspects of geometric approximation via a coupling approach toStein's method. Firstly, we refine precision and increase scope forapplications by convoluting the approximating geometric distribution with asimple translation selected based on the problem at hand. Secondly, we applyour geometric and geometric-type approximations to risk processes and relatedareas, including the approximation of Poisson processes with a random timehorizon and Markov chain hitting times. Particular attention is given togeometric approximation of random sums, for which explicit bounds areestablished. These are applied to give simple approximations, including errorbounds, for the infinite-horizon ruin probability in the compound binomial riskprocess.

On geometric-type approximations with applications to risk and related processes