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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
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References:

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