Total variation error bounds for geometric approximation

We develop a new formulation of Stein’s method to obtain computable upper bounds on the total variation distance between the geometric distribution and a distribution of interest. Our framework reduces the problem to the construction of a coupling between the original distribution and the “discrete equilibrium” distribution from renewal theory. We illustrate the approach in four nontrivial examples: the geometric sum of independent, non-negative, integer-valued random variables having common mean, the generation size of the critical Galton-Watson process conditioned on non-extinction, the indegree of a randomly chosen node in the uniform attachment random graph model, and the total degree of both a fixed and randomly chosen node in the preferential attachment random graph model. In the first two examples we obtain error bounds in a metric that is stronger than those available in the literature, and in the final two examples we provide the first explicit bounds.