Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-30T23:01:37.676Z Has data issue: false hasContentIssue false
  • English
  • Français

Weighted renewal functions: a hierarchical approach

Part of: Distribution theory - Probability Special processes Real functions

Published online by Cambridge University Press:  01 July 2016

Edward Omey  and
Jef L. Teugels
Edward Omey*
Affiliation:
Economische Hogeschool Sint-Aloysius
Jef L. Teugels*
Affiliation:
Katholieke Universiteit Leuven
*
∗ Postal address: Department of Mathematics and Statistics, Economische Hogeschool Sint-Aloysius, Stormstraat 2, 1000-Brussels, Belgium.
∗∗ Postal address: Katholieke Universiteit Leuven, Universitair Centrum Voor Statisiek, De Coylaan 52B, B3001 Heverlee, Belgium. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend classical renewal theorems to the weighted case. A hierarchical chain of successive sharpenings of asymptotic statements on the weighted renewal functions is obtained by imposing stronger conditions on the weighting coefficients.

Keywords

Renewal theoryelementary renewal theoremBlackwell's theoremkey renewal theoremdirect Riemann integrabilityremainder theoremsrenewal moments

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60E05: Distributions 60K10: Applications (reliability, demand theory, etc.) 26A12: Rate of growth of functions, orders of infinity, slowly varying functions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 2 , June 2002 , pp. 394 - 415
Copyright
Copyright © Applied Probability Trust 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A-Hameed, M. S. and Proschan, F. (1973). Nonstationary shock models. Stoch. Process. Appl. 1, 383–404.Google Scholar
Alsmeyer, G. (1991). Erneuerungstheorie. Teubner, Stuttgart.CrossRefGoogle Scholar
Alsmeyer, G. (1991). Some relations between harmonic renewal measures and certain first passage times. Statist. Prob. Lett. 12, 19–27.Google Scholar
Alsmeyer, G. (1992). On generalized renewal measures and certain first passage times. Ann. Prob. 20, 1229–1247.Google Scholar
Baltrūnas, A. and Omey, E. (2001). Second order subexponential sequences and the asymptotic behaviour of their De Pril transform. Liet. Mat. Rink. 41, 21–35.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press.Google Scholar
Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.Google Scholar
Chover, J., Ney, P. and Wainger, S. (1973). Functions of probability measures. J. Anal. Math. 26, 255–302.Google Scholar
Embrechts, P. and Omey, E. (1984). A property of longtailed distributions. J. Appl. Prob. 21, 80–87.Google Scholar
Embrechts, P., Maejima, M. and Omey, E. (1984). A renewal theorem of Blackwell type. Ann. Prob. 12, 561–570.Google Scholar
Embrechts, P., Maejima, M. and Omey, E. (1985). Some limit theorems for generalized renewal measures. J. London Math. Soc. 31, 184–192.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Greenwood, P., Omey, E. and Teugels, J. L. (1982). Harmonic renewal measures. Z. Wahrscheinlich-keitsth. 59, 391–409.Google Scholar
Grübel, R., (1983). Functions of discrete probability measures: rates of convergence in the renewal theorem. Z. Wahrscheinlichkeitsth. 64, 341–357.Google Scholar
Grübel, R., (1986). On harmonic renewal measures. Prob. Theory Relat. Fields 71, 393–404.Google Scholar
Grübel, R., (1987). On subordinated distributions and generalized renewal measures. Ann. Prob. 15, 394–415.CrossRefGoogle Scholar
Grübel, R., (1988). Harmonic renewal measures and the first positive sum. J. London Math. Soc. 38, 179–192.CrossRefGoogle Scholar
Heyde, C. C. and Rohatgi, V. K. (1967). A pair of complementary theorems on convergence rates in the law of large numbers. Proc. Camb. Phil. Soc. 63, 73–82.Google Scholar
Hinderer, K. (1987). Remarks on directly Riemann integrable functions. Math. Nachr. 130, 225–230.Google Scholar
Kalashnikov, V. (1997). Geometric Sums: Bounds for Rare Events with Applications (Math. Appl. 413). Kluwer, Dordrecht.Google Scholar
Kalma, J. M. (1972). Generalized renewal measures. , Groningen University.Google Scholar
Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Keilson, J. (1965). Green's Function Methods in Probability Theory. Hafner, New York.Google Scholar
Lai, T. L. (1975). On uniform integrability in renewal theory. Bull. Inst. Math. Acad. Sinica 3, 99–105.Google Scholar
Nagaev, S. V. (1968). Some renewal theorems. Teor. Veroyat. Primen. 13, 585–601.Google Scholar
Omey, E. (1995). On a subclass of regularly varying functions. J. Statist. Planning Infer. 45, 275–290.CrossRefGoogle Scholar
Omey, E. and Teugels, J. L. (2000). A note on Blackwell's theorem. Submitted.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. L. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.CrossRefGoogle Scholar
Ross, S. M. (1983). Stochastic Processes. John Wiley, New York.Google Scholar
Sgibnev, M. S. (1988). Asymptotic behaviour of higher renewal epochs. Theory Prob. Appl. 36, 508–518.Google Scholar
Sgibnev, M. S. (1997). Asymptotics of the generalized renewal functions when the variance is finite. Theory Prob. Appl. 42, 536–541.Google Scholar
Sgibnev, M. S. (2001). Asymptotic behavior of a generalized renewal measure under weak moment conditions. Siberian Adv. Math. 11, 100–120.Google Scholar
Stam, A. J. (1973). Regular variation of the tail of a subordinated probability distribution. Adv. Appl. Prob. 5, 308–327.Google Scholar