The probability that a stochastic process with negative drift exceed
a value a often has a renewal-theoretic approximation as $a \to \infty$.
Except for a process of iid random variables, this approximation involves a
constant which is not amenable to analytic calculation. Naive simulation of
this constant has the drawback of necessitating a choice of finite a,
thereby hurting assessment of the precision of a Monte Carlo simulation
estimate, as the effect of the discrepancy between a and $\infty$ is
usually difficult to evaluate.
¶ Here we suggest a new way of representing the constant. Our approach
enables simulation of the constant with prescribed accuracy. We exemplify our
approach by working out the details of a sequential power one hypothesis
testing problem of whether a sequence of observations is iid standard normal
against the alternative that the sequence is AR(1). Monte Carlo results are
reported.