Independent random cascades on Galton-Watson trees
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- by Gregory A. Burd and Edward C. Waymire PDF
- Proc. Amer. Math. Soc. 128 (2000), 2753-2761 Request permission
Abstract:
Consider an independent random cascade acting on the positive Borel measures defined on the boundary of a Galton-Watson tree. Assuming an offspring distribution with finite moments of all orders, J. Peyrière computed the fine scale structure of an independent random cascade on Galton-Watson trees. In this paper we use developments in the cascade theory to relax and clarify the moment assumptions on the offspring distribution. Moreover a larger class of initial measures is covered and, as a result, it is shown that it is the Hölder exponent of the initial measure which is the critical parameter in the Peyrière theory.References
- J. D. Biggins, Martingale convergence in the branching random walk, J. Appl. Probability 14 (1977), no. 1, 25–37. MR 433619, DOI 10.2307/3213258
- John Hawkes, Trees generated by a simple branching process, J. London Math. Soc. (2) 24 (1981), no. 2, 373–384. MR 631950, DOI 10.1112/jlms/s2-24.2.373
- Jean-Pierre Kahane, Random multiplications, random coverings, multiplicative chaos, Analysis at Urbana, Vol. I (Urbana, IL, 1986–1987) London Math. Soc. Lecture Note Ser., vol. 137, Cambridge Univ. Press, Cambridge, 1989, pp. 196–255. MR 1009176
- J.-P. Kahane and J. Peyrière, Sur certaines martingales de Benoit Mandelbrot, Advances in Math. 22 (1976), no. 2, 131–145. MR 431355, DOI 10.1016/0001-8708(76)90151-1
- Russell Lyons, Robin Pemantle, and Yuval Peres, Ergodic theory on Galton-Watson trees: speed of random walk and dimension of harmonic measure, Ergodic Theory Dynam. Systems 15 (1995), no. 3, 593–619. MR 1336708, DOI 10.1017/S0143385700008543
- Jacques Peyrière, Calculs de dimensions de Hausdorff, Duke Math. J. 44 (1977), no. 3, 591–601. MR 444911
- Edward C. Waymire and Stanley C. Williams, Multiplicative cascades: dimension spectra and dependence, Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), 1995, pp. 589–609. MR 1364911
- Edward C. Waymire and Stanley C. Williams, A cascade decomposition theory with applications to Markov and exchangeable cascades, Trans. Amer. Math. Soc. 348 (1996), no. 2, 585–632. MR 1322959, DOI 10.1090/S0002-9947-96-01500-0
Additional Information
- Gregory A. Burd
- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- Address at time of publication: Marvell Semiconductor, Inc., 645 Almanor Avenue, Sunnyvale, California 94086
- Email: burd@math.washington.edu, gburd@marvell.com
- Edward C. Waymire
- Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331-4605
- MR Author ID: 180975
- Email: waymire@math.orst.edu
- Received by editor(s): May 14, 1998
- Received by editor(s) in revised form: October 8, 1998
- Published electronically: March 1, 2000
- Communicated by: Stanley Sawyer
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2753-2761
- MSC (2000): Primary 60G57, 60G30, 60G42; Secondary 60K35
- DOI: https://doi.org/10.1090/S0002-9939-00-05279-5
- MathSciNet review: 1657774