Hostname: page-component-7bb8b95d7b-s9k8s Total loading time: 0 Render date: 2024-09-18T23:18:40.084Z Has data issue: false hasContentIssue false
  • English
  • Français

Corrected diffusion approximations in certain random walk problems

Published online by Cambridge University Press:  01 July 2016

D. Siegmund
D. Siegmund*
Affiliation:
Stanford University
*
∗Postal address: Department of Statistics, Stanford University, Stanford, CA 94305, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Correction terms are obtained for the diffusion approximation to one- and two-barrier ruin problems in finite and infinite time. The corrections involve moments of ladder-height distributions, and a method is given for calculating them numerically. Examples show that the corrected approximations can be much more accurate than the originals.

Keywords

DIFFUSION APPROXIMATIONHEAVY TRAFFICRANDOM WALKGAMBLER'S RUIN
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 4 , December 1979 , pp. 701 - 719
Copyright
Copyright © Applied Probability Trust 1979 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research partially supported by National Science Foundation Grant MCS 77-16974 and by ONR Contract N00014-77-C-0306.

References

Aroian, L. A. and Robison, D. E. (1969) Direct methods for exact truncated sequential tests of the mean of a normal distribution. Technometrics 11, 661–675.Google Scholar
Von Bahr, B. (1974) Ruin probabilities expressed in terms of ladder height distributions. Scand. Actuarial J. 57, 190–204.CrossRefGoogle Scholar
Borovkov, A. A. (1962) New limit theorems in boundary problems for sums of independent terms. Selected Translations in Mathematical Statistics and Probability 5, 315–372.Google Scholar
Borovkov, A. A. (1964) Some limit theorems in the theory of mass service. Theory Prob. Appl. 9, 550–565.CrossRefGoogle Scholar
van Dobben de Bruyn, C. S. (1968) Cumulative Sum Tests. Griffin, London.Google Scholar
Epstein, B., Patterson, A. A. and Qualls, C. R. (1963) The exact analysis of sequential life tests with particular application to agree plans. Proceedings of the Aerospace Reliability and Maintainability Conference, 284–311.CrossRefGoogle Scholar
Erdös, P. and Kac, M. (1946) On certain limit theorems of the theory of probability. Bull. Amer. Math. Soc. 52, 292–302.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York.Google Scholar
Hinkley, D. V. (1970) Inference about the change point in a sequence of random variables. Biometrika 57, 1–17.CrossRefGoogle Scholar
Ito, K. and McKean, H. P. Jr. (1965) Diffusion Processes and Their Sample Paths. Springer-Verlag, Berlin.Google Scholar
Lai, T. L. (1976) Asymptotic moments of random walks with applications to ladder variables and renewal theory. Ann. Prob. 4, 51–66.CrossRefGoogle Scholar
Siegmund, D. (1975a) Error probabilities and average sample number of the sequential probability ratio test. J. R. Statist Soc. B 37, 394–401.Google Scholar
Siegmund, D. (1975b) The time until ruin in collective risk theory. Mitt. Verein. Schweiz. Versich.-Math. 75, 157–166.Google Scholar
Siegmund, D. (1978) Estimation following sequential tests. Biometrika 65, 341–349.CrossRefGoogle Scholar
Stone, C. J. (1965) On moment generating functions and renewal theory. Ann. Math. Statist. 36, 1298–1301.CrossRefGoogle Scholar
Woodroofe, M. (1978) Repeated likelihood ratio tests. Technical Report, University of Michigan.Google Scholar