The 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

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

Permanent link to this document
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

Keywords
Overshoot sequential test time series

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


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