## The Annals of Applied Probability

### A new representation for a renewal-theoretic constant appearing in asymptotic approximations of large deviations

#### Abstract

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.

#### Article information

Source
Ann. Appl. Probab., Volume 8, Number 3 (1998), 749-774.

Dates
First available in Project Euclid: 9 August 2002

https://projecteuclid.org/euclid.aoap/1028903449

Digital Object Identifier
doi:10.1214/aoap/1028903449

Mathematical Reviews number (MathSciNet)
MR1627779

Zentralblatt MATH identifier
0937.60082

Subjects
Primary: 60K05: Renewal theory
Secondary: 62L10: Sequential analysis

#### Citation

Yakir, Benjamin; Pollak, Moshe. A new representation for a renewal-theoretic constant appearing in asymptotic approximations of large deviations. Ann. Appl. Probab. 8 (1998), no. 3, 749--774. doi:10.1214/aoap/1028903449. https://projecteuclid.org/euclid.aoap/1028903449

#### References

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