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
Association and random measures
Download PDF
Download PDF
  • Published: March 1990

Association and random measures

  • Steven N. Evans1 

Probability Theory and Related Fields volume 86, pages 1–19 (1990)Cite this article

  • 154 Accesses

  • 20 Citations

  • Metrics details

Summary

Our point of departure is the result, due to Burton and Waymire, that every infinitely divisible random measure has the property variously known as association, positive correlations, or the FKG property. This leads into a study of stationary, associated random measures onR d. We establish simple necessary and sufficient conditions for ergodicity and mixing when second moments are present. We also study the second moment condition that is usually referrent to as finite susceptibility. As one consequence of these results, we can easily rederive some central limit theorems of Burton and Waymire. Using association techniques, we obtain a law of the iterated logarithm for infinitely divisible random measures under simple moment hypotheses. Finally, we apply these results to a class of stationary random measures related to measure-valued Markov branching processes.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  • Birkel, T.: On the convergence rate in the central limit theorem for associated processes. Ann. Probab.16, 1685–1698 (1988)

    Google Scholar 

  • Burton, R., Waymire, E.: Scaling limits for associated random measures. Ann. Probab.13, 1267–1278 (1985)

    Google Scholar 

  • Burton, R., Waymire, E.: The central limit problem for infinitely divisible random measures. In: Taqqu, M., Eberlein, E. (eds.) Dependence in probability and statistics. Boston: Birkhauser 1986

    Google Scholar 

  • Dabrowski, A.R.: A functional law of the iterated logarithm for associated sequences. Statist. Probab. Lett.3, 209–212 (1985)

    Google Scholar 

  • Dawson, D.A., The critical measure diffusion process. Z. Wahrscheinlichkeitstheor. Verw. Geb.40, 125–145 (1977)

    Google Scholar 

  • Dawson, D.A., Ivanoff, G.: Branching diffusions and random measures. In: Joffe, A., Ney, P. (eds.) Advances in probability and related topics, vol. 5, pp. 61–103. New York: Dekker 1978

    Google Scholar 

  • Dynkin, E.B.: Representation of functionals of superprocesses by multiple stochastic integrals, with applications to self-intersection local times. In: Colloque Paul Lévy sur les processus stochastiques, Astérisque 157–158 (Société Mathématique de France) (1988)

  • Esary, J., Proschan, F., Walkup, D.: Association of random variables with applications. Ann. Math. Statist.38, 1466–1474 (1967)

    Google Scholar 

  • Evans, S.N.: Superprocesses and association. Preprint (1989)

  • Evans, S.N.: Rescaling the vacancy of a Boolean coverage process. In: Seminar on stochastic processes, 1989. Boston: Birkhauser 1990

    Google Scholar 

  • Feller, W.: An introduction to probability theory and its applications, vol. II. New York: Wiley 1971

    Google Scholar 

  • Fitzsimmons, P.J.: Construction and regularity of measure-valued Markov branching processes. Isr. J. Math.64, 337–361 (1988)

    Google Scholar 

  • Fortuin, C.M., Kasteleyn, P.W., Ginibre, J.: Correlation inequalities on some partially ordered spaces. Commun. Math. Phys.22, 89–103 (1971)

    Google Scholar 

  • Harris, T.E.: A correlation inequality for Markov processes in partially ordered state spaces. Ann. Probab.5, 451–454 (1977)

    Google Scholar 

  • Herbst, I.W., Pitt, L.D.: Diffusion equation techniques in stochastic monotonicity and positive correlations. Probab. Th. Rel. Fields (to appear)

  • Iscoe, I.: A weighted occupation time for a class of measure-valued critical branching Brownian motion. Probab. Th. Rel. Fields71, 85–116 (1986)

    Google Scholar 

  • Kallenberg, O.: Random measures. Berlin: Akademie-Verlag and New York: Academic Press 1983

    Google Scholar 

  • Lindqvist, B.H.: Association of probability measures on partially ordered spaces. J. Multivar. Anal.26, 111–132 (1988)

    Google Scholar 

  • Matthes, K., Kerstan, J., Mecke, J.: Infinitely divisible point processes. New York: Wiley 1978

    Google Scholar 

  • Newman, C.M.: Normal fluctuations and the FKG inequalities. Commun. Math. Phys.74, 119–128 (1980)

    Google Scholar 

  • Newman, C.M.: Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong, Y.L. (ed.) Inequalities in statistics and probability, pp. 127–140. Hayward: IMS

  • Newman, C.M., Wright, A.L.: An invariance principle for certain dependent sequences. Ann. Probab.9, 671–675 (1981)

    Google Scholar 

  • Petersen, K.: Ergodic theory. Cambridge: Cambridge University Press 1983

    Google Scholar 

  • Spitzer, F.: Principles of random walk, 2nd edn. Berlin Heidelberg New York: Springer 1976

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, University of California at Berkeley, 367 Evans Hall, 94720, Berkeley, CA, USA

    Steven N. Evans

Authors
  1. Steven N. Evans
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported in part by NSF Grant DMS-8701212 at the University of Virginia

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Evans, S.N. Association and random measures. Probab. Th. Rel. Fields 86, 1–19 (1990). https://doi.org/10.1007/BF01207510

Download citation

  • Received: 17 May 1989

  • Revised: 05 January 1990

  • Issue Date: March 1990

  • DOI: https://doi.org/10.1007/BF01207510

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
  • Limit Theorem
  • Mathematical Biology
  • Central Limit
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