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Renewal theory in two dimensions: bounds on the renewal function

Published online by Cambridge University Press:  01 July 2016

Jeffrey J. Hunter
Jeffrey J. Hunter*
Affiliation:
University of Auckland
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Abstract

In two earlier papers [6], [7] the properties of bivariate renewal processes and their associated two-dimensional renewal functions, H(x, y) were examined. By utilising the Fréchet bounds for joint distributions and the properties of univariate renewal processes, a collection of upper and lower bounds for H(x, y) are constructed. The evaluation of these bounds is carried out for the case of the family of bivariate Poisson processes. An interesting by-product of this investigation leads to a new inequality for the median of a Poisson random variable.

Keywords

TWO-DIMENSIONAL RENEWAL THEORYRENEWAL FUNCTIONSBIVARIATE DISTRIBUTION FUNCTIONSBIVARIATE POISSON PROCESSESPOISSON DISTRIBUTION
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 3 , September 1977 , pp. 527 - 541
Copyright
Copyright © Applied Probability Trust 1977 

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References

[1] Cheng, T. T. (1949) The normal approximation to the Poisson distribution and a proof of a conjecture of Ramanujan. Bull. Amer. Math. Soc. 55, 396–401.Google Scholar
[2] Feller, W. (1968) An Introduction to Probability Theory and its Applications Vol. 1, 3rd edn. Wiley, New York.Google Scholar
[3] Feller, W. (1971) An introduction to Probability Theory and its Applications Vol. II, 2nd edn. Wiley, New York.Google Scholar
[4] Fréchet, M. (1951) Sur les tableaux de corrélation dont les marges sont données. Ann. Univ. Lyon A (3) 14, 53–77.Google Scholar
[5] Hoeffding, W. (1940) Masstabinvariante Korrelationstheorie. Schriften Math. Inst. Univ. Berlin. 5, 181–233.Google Scholar
[6] Hunter, J. J. (1974) Renewal theory in two dimensions: Basic results. Adv. Appl. Prob. 6, 376–391.Google Scholar
[7] Hunter, J. J. (1974) Renewal theory in two dimensions: Asymptotic results. Adv. Appl. Prob. 6, 546–562.Google Scholar
[8] Teicher, H. (1955) An inequality on Poisson probabilities. Ann. Math. Statist. 26, 147–149.Google Scholar