Annals of Applied Probability

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

Moshe Pollak and Benjamin Yakir

Full-text: Open access


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

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

First available in Project Euclid: 9 August 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Overshoot sequential test time series


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.

Export citation


  • 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.