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.


60K05 Renewal theory
62L10 Sequential statistical analysis
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[1] ANDERSON, T. W. 1971. The Statistical Analy sis of Time Series. Wiley, New York. Z.
[2] HOGAN, M. and SIEGMUND, D. 1986. Large derivations for the maxima of some random fields. Adv. in Appl. Math. 7 2 22. Z. · Zbl 0612.60029
[3] KESTEN, H. 1974. Renewal theory for functionals of a Markov Chain with general state space. Ann. Probab. 2 355 386. Z. · Zbl 0303.60090
[4] LALLEY, S. P. 1986. Renewal theorem for a class of stationary sequences. Probab. Theory Related Fields 72 195 213. Z. · Zbl 0597.60083
[5] SIEGMUND, D. 1976. Importance sampling in the Monte Carlo study of sequential tests. Ann. Statist. 4 673 684. Z. · Zbl 0353.62044
[6] SIEGMUND, F. 1985. Sequential Analy sis: Tests and Confidence Intervals. Springer, New York. Ź. · Zbl 0573.62071
[7] VILLE, J. 1939. Etude Critique de la Notion de Collectif. Gauthier-Villars, Paris. Z. · Zbl 0021.14601
[8] YAKIR, B. 1995. A note on the run length to false alarm of a change-point detection policy. Ann. Statist. 23 272 281. · Zbl 0828.62072
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