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Fixed points of the smoothing transformation

Publication ,  Journal Article
Durrett, R; Liggett, TM
Published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
September 1, 1983

Let W 1,..., W N be N nonnegative random variables and let {Mathematical expression} be the class of all probability measures on [0, ∞). Define a transformation T on {Mathematical expression} by letting Tμ 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. © 1983 Springer-Verlag.

Duke Scholars

Published In

Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

DOI

EISSN

1432-2064

ISSN

0044-3719

Publication Date

September 1, 1983

Volume

64

Issue

3

Start / End Page

275 / 301
 

Citation

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Durrett, R., & Liggett, T. M. (1983). Fixed points of the smoothing transformation. Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete, 64(3), 275–301. https://doi.org/10.1007/BF00532962
Durrett, R., and T. M. Liggett. “Fixed points of the smoothing transformation.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 64, no. 3 (September 1, 1983): 275–301. https://doi.org/10.1007/BF00532962.
Durrett R, Liggett TM. Fixed points of the smoothing transformation. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 1983 Sep 1;64(3):275–301.
Durrett, R., and T. M. Liggett. “Fixed points of the smoothing transformation.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete, vol. 64, no. 3, Sept. 1983, pp. 275–301. Scopus, doi:10.1007/BF00532962.
Durrett R, Liggett TM. Fixed points of the smoothing transformation. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 1983 Sep 1;64(3):275–301.
Journal cover image

Published In

Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

DOI

EISSN

1432-2064

ISSN

0044-3719

Publication Date

September 1, 1983

Volume

64

Issue

3

Start / End Page

275 / 301