Fixed points of the smoothing transformation

  • Richard Durrett
  • Thomas M. Liggett


Let W 1,..., W N 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 W 1X1+ ... + W N X N , where the X i are independent random variables with distribution μ, which are independent of W 1,..., W N 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 W i 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, EW i γ <∞ 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.


Random Walk Invariant Measure Independent Random Variable Unique Fixed Point Nonnegative Random Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

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

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