Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T12:03:00.944Z Has data issue: false hasContentIssue false
  • English
  • Français

Renewal theory in two dimensions: Basic results

Published online by Cambridge University Press:  01 July 2016

Jeffrey J. Hunter
Jeffrey J. Hunter*
Affiliation:
University of Auckland, New Zealand
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper a unified theory for studying renewal processes in two dimensions is developed. Bivariate generating functions and bivariate Laplace transforms are the basic tools used in generalizing the standard theory of univariate renewal processes. An example involving a bivariate exponential distribution is presented. This is used to illustrate the general theory and explicit expressions for the two-dimensional renewal density, the two-dimensional renewal function, the correlation between the marginal univariate renewal counting processes, and other related quantities are derived.

Keywords

TWO-DIMENSIONAL RENEWAL THEORYBIVARIATE GENERATING FUNCTIONSBIVARIATE LAPLACE TRANSFORMSRENEWAL FUNCTIONSRENEWAL DENSITIESBIVARIATE EXPONENTIAL DISTRIBUTION
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 2 , June 1974 , pp. 376 - 391
Copyright
Copyright © Applied Probability Trust 1974 

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.)

References

[1] Cox, D. R. (1962) Renewal Theory. Methuen, London.Google Scholar
[2] Ditkin, V. A. and Prudnikov, A. P. (1962) Operational Calculus in Two Variables and its Applications. Translated by Wishart, D. M. G.. Pergamon, London.Google Scholar
[3] Downton, F. (1970) Bivariate exponential distributions in reliability theory. J. R. Statist. Soc. B 32, 408–417.Google Scholar
[4] Feller, W. (1968) An Introduction to Probability Theory and its Applications. Vol. I, 3rd. ed. Wiley, New York.Google Scholar
[5] Hunter, J. J. (1973) Renewal theory in two dimensions: Basic results. Institute of Statistics Mimeo Series No. 861, University of North Carolina at Chapel Hill.Google Scholar
[6] Johnson, N. L. and Kotz, S. (1972) Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York.Google Scholar
[7] Kibble, W. F. (1941) A two variate gamma-type distribution. Sankhyā 5, 137–150.Google Scholar
[8] Nagao, M. and Kadoya, M. (1971) Two-variate exponential distribution and its numerical table for engineering application. Bull. Disas. Prev. Res. Inst. Kyoto Univ. 20, 183–215.Google Scholar
[9] Ramasubban, T. A. (1958) The mean difference and mean deviation of some discontinuous distributions. Biometrika 45, 549–556.Google Scholar
[10] Smith, W. L. (1958) Renewal theory and its ramifications. J. R. Statist. Soc. B 20, 243–302.Google Scholar
[11] Voelker, D. and Doetsch, G. (1950) Die Zweidimensionale Laplace-transformation. Verlag Birkhäuser, Basel.Google Scholar