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
On the domain of attraction of an operator between supremum and sum
Download PDF
Download PDF
  • Published: June 1991

On the domain of attraction of an operator between supremum and sum

  • Priscilla E. Greenwood1 &
  • Gerard Hooghiemstra2 

Probability Theory and Related Fields volume 89, pages 201–210 (1991)Cite this article

  • 96 Accesses

  • 19 Citations

  • Metrics details

Summary

Between the operations which produce partial maxima and partial sums of a sequenceY 1,Y 2, ..., lies the inductive operation:X n =X n-1∨(αX n-1+Y n ),n≧1, for 0<α<1. If theY n are independent random variables with common distributionF, we show that the limiting behavior of normed sequences formed from {X n ,n≧1}, is, for 0<α<1, parallel to the extreme value case α=0. ForF∈D(Φγ) we give a full proof of the convergence, whereas forF∈D(Ψγ)∪D(Λ), we only succeeded in proving tightness of the involved sequence. The processX n is interesting for some applied probability models.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Daley, D.J., Haslett, J.: A thermal energy storage process with controlled input. Adv. Appl. Probab.14, 257–271 (1982)

    Google Scholar 

  2. Geluk, J.L., Haan, L. de: Regular variation, extensions and tauberian theorems. CWI Tract40, CWI, Amsterdam 1987

    Google Scholar 

  3. Gnedenko, B.: Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math.44, 423–453 (1943)

    Google Scholar 

  4. Gnedenke, B., Kolmogorov, A.N.: Limit distributions for sums of independent random variables. Reading, Mass: Addison Wesley 1968

    Google Scholar 

  5. Greenwood, P.E., Hooghiemstra, G.: An extreme-type limit law for a storage process. Math. Oper. Res.13, 232–242 (1988)

    Google Scholar 

  6. Haan, L. de: On regular variation and its application to the weak convergence of sample extremes. Mathematical Centre Tracts32, CWI, Amsterdam 1970

    Google Scholar 

  7. Haslett, J.: Problems in the storage of solar thermal energy. In: Jacobs, O.L.R., et al. (eds.) Analysis and optimization of stochastic systems. London: Academic Press 1980

    Google Scholar 

  8. Haslett, J.: New bounds for the thermal energy storage process with stationary input. J. Appl. Probab.19, 894–899 (1982)

    Google Scholar 

  9. Hooghiemstra, G., Keane, M.: Calculation of the equilibrium distribution for a solar energy storage model. J. Appl. Probab.22, 852–864 (1985)

    Google Scholar 

  10. Hooghiemstra, G., Scheffer, C.L.: Some limit theorems for an energy storage model. Stochastic Processes Appl.22, 121–128 (1986)

    Google Scholar 

  11. Lamperti, J.: Serni-stable Markov processe, I. Z. Wahrscheinlixhkeitstheor. Verw. Geb.22, 205–225 (1972)

    Google Scholar 

  12. Resnick, S.I.: Pomt processes, regular variation and weak convergence. Adv. Appl. Probab.18, 66–138 (1986)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, University of British Columbia, 121-1984 Mathematics Road, V6T 1Y4, Vancouver, B.C., Canada

    Priscilla E. Greenwood

  2. Faculty of Mathematics and Informatics, Department of Statistics, Probability and Operations Research, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands

    Gerard Hooghiemstra

Authors
  1. Priscilla E. Greenwood
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gerard Hooghiemstra
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Greenwood, P.E., Hooghiemstra, G. On the domain of attraction of an operator between supremum and sum. Probab. Th. Rel. Fields 89, 201–210 (1991). https://doi.org/10.1007/BF01366906

Download citation

  • Received: 19 September 1990

  • Revised: 26 March 1991

  • Issue Date: June 1991

  • DOI: https://doi.org/10.1007/BF01366906

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

  • Stochastic Process
  • Probability Theory
  • Probability Model
  • Mathematical Biology
  • Independent Random Variable
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