## A new representation for a renewal-theoretic constant appearing in asymptotic approximations of large deviations.(English)Zbl 0937.60082

Authors’ summary: The probability that a stochastic process with negative drift exceeds 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 autoregressive with $$p=1$$ and a known autoregression parameter. Monte Carlo results are reported.

### MSC:

 60K05 Renewal theory 62L10 Sequential statistical analysis

### Keywords:

overshoot; sequential test; time series
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### References:

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