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Transactions of the American Mathematical Society

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A cascade decomposition theory with applications to Markov and exchangeable cascades

Authors: Edward C. Waymire and Stanley C. Williams
Journal: Trans. Amer. Math. Soc. 348 (1996), 585-632
MSC (1991): Primary 60G57, 60G30, 60G42; Secondary 60K35, 60D05, 60J10, 60G09
MathSciNet review: 1322959
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Abstract: A multiplicative random cascade refers to a positive $T$-martingale in the sense of Kahane on the ultrametric space $T = { \{ 0,1,\dots ,b-1 \} }^{\mathbf{N}}.$ A new approach to the study of multiplicative cascades is introduced. The methods apply broadly to the problems of: (i) non-degeneracy criterion, (ii) dimension spectra of carrying sets, and (iii) divergence of moments criterion. Specific applications are given to cascades generated by Markov and exchangeable processes, as well as to homogeneous independent cascades.

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Additional Information

Edward C. Waymire

Stanley C. Williams

Keywords: Martingale, Hausdorff dimension, tree, cascade, random measure, percolation, exchangeable
Received by editor(s): August 18, 1994
Additional Notes: The authors would like to thank an anonymous referee for several suggestions, both technical and otherwise, which improved the readability of this paper. This research was partially supported by grants from NSF and NASA
Article copyright: © Copyright 1996 American Mathematical Society