Fixed points of the smoothing transformation

  • Richard Durrett
  • Thomas M. Liggett
Article

Summary

Let W1,..., WN be N nonnegative random variables and let \(\mathfrak{M}\) be the class of all probability measures on [0, ∞). Define a transformation T on \(\mathfrak{M}\) by letting be the distribution of W1X1+ ... + WNXN, where the Xi are independent random variables with distribution μ, which are independent of W1,..., WN as well. In earlier work, first Kahane and Peyriere, and then Holley and Liggett, obtained necessary and sufficient conditions for T to have a nontrivial fixed point of finite mean in the special cases that the Wi are independent and identically distributed, or are fixed multiples of one random variable. In this paper we study the transformation in general. Assuming only that for some γ>1, EWiγ<∞ for all i, we determine exactly when T has a nontrivial fixed point (of finite or infinite mean). When it does, we find all fixed points and prove a convergence result. In particular, it turns out that in the previously considered cases, T always has a nontrivial fixed point. Our results were motivated by a number of open problems in infinite particle systems. The basic question is: in those cases in which an infinite particle system has no invariant measures of finite mean, does it have invariant measures of infinite mean? Our results suggest possible answers to this question for the generalized potlatch and smoothing processes studied by Holley and Liggett.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dawson, D.A.: The critical measure diffusion process. Z. Wahrscheinlichkeitstheorie verw. Gebiete 40, 125–145 (1977)MATHCrossRefGoogle Scholar
  2. 2.
    Durrett, R.: An infinite particle system with additive interactions. Advances in Appl. Probability 11, 355–383 (1979)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Feller, W.: An introduction to probability theory and its applications, Vol. II. New York: J. Wiley and Sons 1966MATHGoogle Scholar
  4. 4.
    Harris, T.E.: A correlation inequality for Markov processes in partially ordered state spaces. Ann. Probability 5, 451–454 (1977)MATHMathSciNetGoogle Scholar
  5. 5.
    Holley, R., Liggett, T.M.: Generalized potlatch and smoothing processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 55, 165–195 (1981)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Kahane, J.P., Peyriere, J.: Sur certaines martingales de Benoit Mandelbrot. Advances in Math. 22, 131–145 (1976)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Kallenberg, O.: Stability of critical cluster fields. Math. Nachr. 77, 7–43 (1977)MATHMathSciNetGoogle Scholar
  8. 8.
    Kerstan, J., Matthes, K., Mecke, J.: Infinitely divisible point processes. New York: J. Wiley and Sons 1978MATHGoogle Scholar
  9. 9.
    Kingman, J.F.C.: The first birth problem for an age-dependent branching process. Ann. Probability 3, 790–801 (1975)MATHMathSciNetGoogle Scholar
  10. 10.
    Liggett, T.M.: Random invariant measures for Markov chains, and independent particle systems. Z. Wahrscheinlichkeitstheorie verw. Gebiete 45, 297–313 (1978)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Liggett, T.M., Spitzer, F.: Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrscheinlichkeitstheorie verw. Gebiete 56, 443–468 (1981)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Port, S.C., Stone, C.J.: Potential theory of random walky on abelian groups. Acta Math. 122, 19–114 (1969)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Royden, H.: Real Analysis, second edition. New York: MacMillan 1968MATHGoogle Scholar
  14. 14.
    Spitzer, F.: Infinite systems with locally interacting components. Ann. Probability 9, 349–364 (1981)MATHMathSciNetGoogle Scholar
  15. 15.
    Stone, C.J.: On a theorem of Dobrushin. Ann. Math. Statist. 39, 1391–1401 (1968)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Richard Durrett
    • 1
  • Thomas M. Liggett
    • 1
  1. 1.Mathematics DepartmentUniversity of CaliforniaLos AngelesUSA

Personalised recommendations