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Renewal theory in two dimensions: asymptotic results

Published online by Cambridge University Press:  01 July 2016

Jeffrey J. Hunter
Jeffrey J. Hunter*
Affiliation:
University of Auckland, New Zealand
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Abstract

In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (Nx(1), Ny(2)), the distribution of the two-dimensional renewal counting process Nx,y and the two-dimensional renewal function &Nx,y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended.

Keywords

TWO-DIMENSIONAL RENEWAL THEORYRENEWAL FUNCTIONSASYMPTOTIC DISTRIBUTIONSMINIMUM OF TWO CORRELATED NORMAL RANDOM VARIABLES
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 3 , September 1974 , pp. 546 - 562
Copyright
Copyright © Applied Probability Trust 1974 

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References

Bibliography on multi-dimensional renewal theory

[1] Bickel, P. J. and Yahav, J. A. (1965) Renewal theory in the plane. Ann. Math. Statist. 36, 946–955. MR: 31–817.Google Scholar
[2] Bickel, P. J. and Yahav, J. A. (1965) The number of visits of vector walks to bounded regions. Israel J. Math. 3, 181–186. MR: 34–871.Google Scholar
[3] Chung, K. L. (1952) On the renewal theorem in higher dimensions. Skand. Aktuarietidskr. 35, 188–194. MR: 14–994.Google Scholar
[4] Doney, R. A. (1966) An analogue of the renewal theorem in higher dimensions. Proc. London Math. Soc. 16, 669–684. MR: 34–3673.Google Scholar
[5] Farrell, R. H. (1964) Limit theorems for stopped random walks. Ann. Math. Statist. 35, 1332–1343. MR: 29–2868.CrossRefGoogle Scholar
[6] Farrell, R. H. (1966) Limit theorems for stopped random walks II. Ann. Math. Statist. 37, 860–865. MR: 34–2080.Google Scholar
[7] Farrell, R. H. (1966) Limit theorems for stopped random walks III. Ann. Math. Statist. 37, 1510–1527. MR: 37–5944.Google Scholar
[8] Hunter, J.J. (1974) Renewal theory in two dimensions: Basic results. Adv. Appl. Prob. 6, 376–391.Google Scholar
[9] Kennedy, D. P. (1971) A functional central limit theorem for k-dimensional renewal theory. Ann. Math. Statist. 42, 376–380. MR: 43–5589.Google Scholar
[10] Mode, C. J. (1967) A renewal density theorem in the multidimensional case. J. Appl. Prob. 4, 62–76. MR: 34–8460.CrossRefGoogle Scholar
[11] Ney, P. and Spitzer, F. (1966) The Martin boundary for random walk. Trans. Amer. Math. Soc. 121, 116–132. MR: 33–3354.Google Scholar
[12] Smith, W. L. (1958) Renewal theory and its ramifications. J. R. Statist. Soc. B 20, 243–302. MR: 20–5534.Google Scholar
[13] Stam, A. J. (1968) Two theorems in r-dimensional renewal theory. Z. Wahrscheinlichkeitsth. 10, 81–86. MR: 38–3930.Google Scholar
[14] Stam, A. J. (1969) Renewal theory in r-dimensions, I. Compositio Math. 21, 383–399. MR: 41–6327.Google Scholar
[15] Stam, A. J. (1971) Renewal theory in r-dimensions, II. Compositio Math. 23, 1–13. MR: 43–6994.Google Scholar
[16] Stam, A. J. (1971) Local central limit theorem for first entrance of a random walk into a half space. Compositio Math. 23, 15–23. MR: 43–8131.Google Scholar

Additional references

[17] Afonja, B. (1972) The moments of the maximum of correlated normal and t-variates. J. R. Statist. Soc. B 34, 251–262.Google Scholar
[18] Feller, W. (1949) Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67, 98–119.Google Scholar
[19] Mann, H. B. and Wald, A. (1943) On stochastic limit and order relationships. Ann. Math. Stastist. 14, 217–226.Google Scholar
[20] Takács, L. (1956) On a probability theorem arising in the theory of counters. Proc. Camb. Phil. Soc. 52, 488–498.Google Scholar