Second-Order Renewal Theorem in the Finite-Means Case
Abstract
Let F be a distribution function (d.f.) on $(0, \infty )$ and let~U be the renewal function associated with F. If F has a finite first moment~$\mu$, then it is well known that $U(t)$ asymptotically equals $t/\mu$. It is also well known that $U(t)-t/\mu $ asymptotically behaves as $S(t)/\mu, $ where~S denotes the integral of the integrated tail distribution~$F_1$ of~F. In this paper we discuss the rate of convergence of $U(t)-t/\mu -S(t)/\mu $ for a large class of distribution functions. The estimate improves earlier results of Geluk, Teugels, and Embrechts and Omey.
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References
1.
A. Baltrūnas and E. Omey, The rate of convergence for subexponential distributions, Liet. Mat. Rink., 38 (1998), pp. 1–18.
2.
A. Baltrūnas, Simple proof of the second‐order renewal theorem, in Proc. Lithuanian Math. Soc., 2 (1998), pp. 489–491.
3.
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, UK, 1987.
4.
P. Embrechts and E. Omey, A property of longtailed distributions, J. Appl. Probab., 21 (1984), pp. 80–87.
5.
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed., Wiley, New York, 1971.
6.
J. B. G. Frenk, On Banach Algebras, Renewal Measures and Regenerative Processes, CWI Tract 38, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1987.
7.
J. L. Geluk, A renewal theorem in the finite‐mean case, Proc. Amer. Math. Soc., 125 (1997), pp. 3407–3413.
8.
P. Ney, A refinement of the coupling method in renewal theory, Stochastic Process. Appl., 11 (1981), pp. 11–26.
9.
E. Omey, On a subclass of regularly varying functions, J. Statist. Plann. Inference, 45 (1995), pp. 275–290.
10.
E. J. G. Pitman, Subexponential distribution functions, J. Austral. Math. Soc. Ser. A, 29 (1980), pp. 337–347.
11.
M. S. Sgibnev, On the renewal theorem in the case of infinite variance, Sibirsk. Mat. Zh., 22 (1981), pp. 178–189 (in Russian).
12.
J. L. Teugels, Renewal theorems when the first or the second moment is infinite, Ann. Math. Statist., 39 (1968), pp. 1210–1219.
13.
J. L. Teugels, The class of subexponential distributions, Ann. Probab., 3 (1975), pp. 1000–1011.
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Published In
Theory of Probability & Its Applications
Pages: 127 - 132
ISSN (online): 1095-7219
Copyright
Copyright © 2003 Society for Industrial and Applied Mathematics.
History
Published online: 25 July 2006
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Cited By
- Intuitive approximations for the renewal functionStatistics & Probability Letters, Vol. 84 | 1 Jan 2014
- Estimating the Renewal Function When the Second Moment Is InfiniteStochastic Models, Vol. 23, No. 1 | 7 Feb 2007
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