Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
A strong renewal theorem for generalized renewal functions in the infinite mean case
Download PDF
Download PDF
  • Published: December 1988

A strong renewal theorem for generalized renewal functions in the infinite mean case

  • Kevin K. Anderson1 &
  • Krishna B. Athreya1 

Probability Theory and Related Fields volume 77, pages 471–479 (1988)Cite this article

Summary

LetF(x) be a nonarithmetic c.d.f. on (0, ∞) such that 1 —F(x)=x −α L(x), whereL(x) is slowly varying and 0≤α≤1. Leta(x) be regularly varying with exponent β≥1. A strong renewal theorem (of Blackwell type) for generalized renewal functions of the form\(G(t) \equiv \sum\limits_{n = 0}^\infty {a(n) F^n (t)} \) is proved here, thus extending the recent work of Embrechts, Maejima and Omey [1] and that of Erickson [4].

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Embrechts, P., Maejima, M., Omey, E.: A renewal theorem of Blackwell type. Ann. Probab.12, 561–570 (1984)

    Google Scholar 

  2. Embrechts, P., Maejima, M., Omey, E.: Some limit theorems for generalized renewal measures. J. London Math. Soc., II. Ser.31, 184–192 (1985)

    Google Scholar 

  3. Embrechts, P., Omey, E.: On subordinated distributions and random record processes. Math. Proc. Camb. Philos. Soc.93, 339–353 (1983)

    Google Scholar 

  4. Erickson, K.B.: Strong renewal theorems with infinite mean. Trans. Am. Math. Soc.151, 263–291 (1970)

    Google Scholar 

  5. Greenwood, P., Omey, E., Teugels, T.L.: Harmonic renewal measures. Z. Wahrscheinlichkeitstheor. verw. Geb.59, 391–409 (1982)

    Google Scholar 

  6. Grübel, R.: On harmonic renewal measures. Probab. Th. Rel. Fields71, 393–404 (1986)

    Google Scholar 

  7. Omey, E.: Multivariate regular variation and its applications in probability theory. Ph.D. dissertation, University of Leuven 1982

  8. Stam, A.: Regular variation of the tail of a subordinated probability distribution. Adv. Appl. Probab.5, 308–327 (1973)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Iowa State University, 50010, Ames, IA, USA

    Kevin K. Anderson & Krishna B. Athreya

Authors
  1. Kevin K. Anderson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Krishna B. Athreya
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Kevin K. Anderson is now at Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 914550 USA. His research was performed in part under the auspices of the U.S. Department of Energy at LLNL under Contract W-7405-Eng-48.

The research of Krishna B. Athreya was supported in part by NSF Grants DMS-8502311 and DMS-8706319.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Anderson, K.K., Athreya, K.B. A strong renewal theorem for generalized renewal functions in the infinite mean case. Probab. Th. Rel. Fields 77, 471–479 (1988). https://doi.org/10.1007/BF00959611

Download citation

  • Received: 01 July 1985

  • Revised: 18 December 1987

  • Issue Date: December 1988

  • DOI: https://doi.org/10.1007/BF00959611

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Recent Work
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Generalize Renewal
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature