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A renewal theorem in the finite-mean case

Author: J. L. Geluk
Journal: Proc. Amer. Math. Soc. 125 (1997), 3407-3413
MSC (1991): Primary 60K05
MathSciNet review: 1403127
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Abstract: Let $F(.)$ be a c.d.f. on $(0,\infty )$ such that $\overline F(.) \equiv 1-F(.)$ is regularly varying with exponent $-\alpha ,~1<\alpha <2$. Then $U(t)- \frac {t}{\mu } -\frac {1}{\mu ^2} \int _0^t \int _s^\infty \overline F(v) dv ds = O(t^4 \overline F(t)^2 \overline F(t^2\overline F(t)))$ as $t \to \infty $, where $U(t)=EN(t)$ is the renewal function associated with $F(t)$. Moreover similar estimates are given for distributions in the domain of attraction of the normal distribution and for the variance of $N(t).$ The estimates improve earlier results of Teugels and Mohan.

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  • 1. Anderson, K.K., Athreya, K.B. (1987). A renewal theorem in the infinite mean case. Ann. Prob. 15 388-393. MR 88f:60154
  • 2. Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular variation. Cambridge Univ. Press. MR 88i:26004
  • 3. Erickson, K.B. (1970). Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151 263-291. MR 42:3873
  • 4. Feller, W. (1949). Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67 98-119. MR 11:255c
  • 5. Feller, W. (1971). An introduction to probability theory and its applications 2, 2nd ed. Wiley, New York. MR 42:5292
  • 6. Frenk, J.B.G. (1983). On renewal theory, Banach Algebras and functions of bounded increase, Centre for Mathematics and Computer Science, Amsterdam.
  • 7. Geluk, J.L., de Haan, L. (1987). Regular variation, extensions and Tauberian Theorems, CWI Tract 40, Centre for Mathematics and Computer Science, Amsterdam. MR 89a:26002
  • 8. Mohan, N.R. (1976). Teugels' renewal theorem and stable laws. Ann. Prob. 4 863-868. MR 54:6312
  • 9. Resnick, S.I. (1987). Extreme values, regular variation and point processes, Springer Verlag, Berlin. MR 89b:60241
  • 10. Smith, W.L. (1954). Asymptotic renewal theorems. Proc. Roy. Soc. Edinburgh Ser. A. 64 9-48. MR 15:722f
  • 11. Smith, W.L. (1962). A note on the renewal function when the mean renewal lifetime is infinite. J. Roy. Statist. Soc. Ser. B 23 230-237. MR 23:A2963
  • 12. Teugels, J.L. (1968). Renewal theorems when the first or the second moment is infinite. Ann. Math. Statist. 39 1210-1219. MR 37:5952

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Additional Information

J. L. Geluk
Affiliation: Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, NL-3000 DR Rotterdam, The Netherlands

Keywords: Renewal function, regular variation, key renewal theorem, domain of attraction
Received by editor(s): March 12, 1996
Received by editor(s) in revised form: June 21, 1996
Communicated by: Stanley Sawyer
Article copyright: © Copyright 1997 American Mathematical Society