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The exact order of normal approximation in bivariate renewal theory

Published online by Cambridge University Press:  01 July 2016

Ibrahim A. Ahmad
Ibrahim A. Ahmad*
Affiliation:
Memphis State University
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Abstract

Equivalence of rates of convergence in the central limit theorem between the vector of maximum sums and the corresponding first-passage variables is established. The bivariate case is studied. Analogous results about the equivalence between the vector of partial sums and corresponding renewal variables are also given and as a consequence we obtain a generalization of a theorem of Hunter (1974). Extension of the main result to more general first-passage times is also developed.

Keywords

RATES OF CONVERGENCECENTRAL LIMIT THEOREMFIRST-PASSAGE TIMESMAXIMUM PARTIAL SUMSRENEWAL VARIABLES
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 1 , March 1981 , pp. 113 - 128
Copyright
Copyright © Applied Probability Trust 1981 

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References

[1] Ahmad, I. A. and Lin, P. E. (1977) A Berry–Esseén type theorem. Utilitas Math. 11, 153–160.Google Scholar
[2] Dunnage, J. E. A. (1970) On Sadikova's method in the central limit theorem. J. London Math. Soc. (2) 2, 49–60.CrossRefGoogle Scholar
[3] Gut, A. (1974) On moments and limit distributions of some first passage times. Ann. Prob. 2, 277–308.Google Scholar
[4] Hunter, J. J. (1974) Renewal theory in two dimensions: asymptotic results. Adv. Appl. Prob. 6, 546–562.CrossRefGoogle Scholar