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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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  • BW Bhattacharya, R.N. and E. Waymire, Stochastic processes with applications, John Wiley and Sons, New York, 1990, MR 91m:60001.
  • CF Chung, K.L. and W.H.J. Fuchs, On the distribution of values of sums of independent random variables, Memoirs of the American Mathematical Society, no. 6, Providence, RI (1951), MR 12:722e.
  • CK Collet,P. and F. Koukiou, Large deviations for multiplicative, Commun. Math. Phys. 147 (1992), 329-342, MR 94a:82003.
  • CO Cohn, D.C., Measure theory, Birkhäuser, Boston (1980), MR 81k:28001.
  • CU Cutler, C., The Hausdorff dimension spectrum of finite measures in Euclidean space,, Can. J. Math. XXXVIII (1986), 1459-1484, MR 88b:28013.
  • DER Derrida, B., Mean field theory of directed polymers in a random medium and beyond, Physica Scripta T38 (1991), 6-12.
  • DERSP Derrida, B. and H. Spohn, Polymers on disordered trees, spin glasses, traveling waves, Jour. Stat. Phys. 51 (1988), 817-840, MR 90i:82045.
  • DP Dawson, D. and E. Perkins, Historical processes, Mem. Amer. Math. Soc., Providence RI (1991), MR 92a:60145.
  • DEUST Deuschel J-D. and D. Stroock, Large deviations, Academic Press, San Diego, CA, 1989, MR 9h:60026.
  • DL Durrett,R. and T. M. Liggett, Fixed points of the smoothing transformation, Z. Wahr. verw. Geb. 64 (1983), 275-301, MR 85e:60059.
  • DUNSCH Dunford N. and J.T. Schwartz, Linear operators, Part 1, Wiley Interscience, NY, 1958, MR 22:8302.
  • DYN Dynkin, E.B., Superdiffusions and parabolic nonlinear differential equations, Ann. Probab. 20 (1992), 942-962, MR 93d:60124.
  • FALC Falconer,K., The multifractal spectrum of statistically self- similar measures, preprint (1993).
  • GW Gupta, V.K. and E. Waymire, A statistical analysis of mesoscale rainfall as a random cascade, Jour. Appld. Meteor. 32 (2) (1993), 251-267.
  • GMW Graf, S., D. Mauldin, and S.C. Williams, The exact Hausdorff dimension in random recursive constructions, Memoirs Am. Math. Soc. 71, no. 381 (1988), MR 88k:28010.
  • GUI Guivarc'h, Y., Sur une extension de la notion de loi semi-stable,, Ann. Inst. Henri Poincare 26 (2) (1990), 261-285, MR 91i:60141.
  • HAR Harris, T., The theory of branching processes, Prentice Hall, Englewood Cliffs, NJ, 1963, MR 29:664.
  • HL Holley, R. and T. Liggett (1981):, Generalized potlatch and smoothing processes, Z. fur Wahr. verw. Geb. 55, 165-195, MR 82i:60176.
  • HW Holley, R. and E. Waymire, Multifractal dimensions and scaling exponents for strongly bounded random cascades, Ann. Applied Probab. 2 (1992), 819-845, MR 93k:60122.
  • KP Kahane, J.P. and J. Peyrière, Sur certaines martingales de Benoit Mandelbrot, Advances in Math. 22 (1976), 131-145, MR 55:4355.
  • K1 Kahane, J.P., Multiplications aleatoires et dimensions de Hausdorff, Ann. Inst. Poin-
    care 23 (1987), 289-296, MR 88h:60100.
  • K2 Kahane, J.P., Positive Martingales and random measures, Chinese Ann. Math. 8b (1987), 1-12, MR 88j:60098.
  • K3 Kahane, J.P., Fractal Geometry and Analysis, ed. J. Belair, S. Dubac, Kluwer Academic Publ., The Netherlands, 1991, MR 94m:60025.
  • K4 Kahane, J.P., Proceedings of the Special Year in Modern Analysis, London Math. Soc. Lect. Notes ed by E. Berkson, N. Tenney Peck, J. Jerry Uhl, vol. 137, Cambridge Univ. Press, 1989, pp. 196-255, MR 91e:60152.
  • KK Kahane, J.P. and Y. Katznelson, Decomposition des measures selon la dimension, Colloq. Math. LVII (1990), 269-279, MR 91g:28018.
  • KES Kesten, H., Subdiffusive behavior of random walk on a random cluster, Ann. Inst. H. Poincare Prob. Stat. 22 (1986), 425-487, MR 88b:60232.
  • KW Kiefer,J. and J. Wolfowitz, On the characteristics of the general queueing process with applications to random walk, Ann. Math. Stat. 27 (1956), 147-161, MR 17:980d.
  • KIF Kifer, Y., Ergodic theory of random transformations, Birkhäuser, Boston, 1986, MR 89c:56069.
  • KOU Koukiou, F., The mean-field theory of spin glass and directed polymer models in random media, preprint (1993).
  • LIND Lindvall, T., Lectures on the coupling method, Wiley, NY, 1992, MR 94c:60002.
  • LY Lyons, R., Random walks and percolation on trees, Ann. Prob. 18 (3) (1990), 931-958, MR 91i:60179.
  • MW Mauldin, D. and S.C. Williams, Random recursive constructions asymptotic geometric and topological properties, Trans. Amer. Math. Soc. 295 (1986), 325-346, MR 87j:60027.
  • OG Over, T. and V. K. Gupta, Statistical analysis of mesoscale rainfall: dependence of a random cascade generator on the large scale forcing, J. Appld. Meteor. 33 (1995), 1526-1542.
  • OR Orey, S., Limit theorems for markov chain transition probabilities, Van Nostrand, NY, 1971, MR 48:3123.
  • P Peyrière, J., Calculs de dimensions de hausdorff, Duke Mathematical Journal 44 (1977), 591-601, MR 56:3257.
  • SH Shepp, L., Covering the circle with random arcs, Israel J. Math. 11 (1972), 328-345, MR 48:1284.
  • SLSL Schmitt, F., D. Lavallee, D. Schertzer, S. Lovejoy, Empirical determination of universal multifractal exponents in turbulent velocity fields, Phys. Rev. Lett. 68 (1992), 305-308, MR 94e:76045.
  • WW1 Waymire, E. and S.C. Williams, Markov cascades, IMA Volume on Branching Processes, ed by K. Athreya and P. Jagers, in press, 1995.
  • WW2 Waymire, E. and S.C. Williams, Multiplicative cascades: dimension spectra and dependence, special issue, Journal of Fourier Analysis and Applications (1995), 589-609.
  • WW3 Waymire, E. and S.C. Williams, Correlated spin glasses on trees, preprint (1995).
  • WW4 Waymire, E. and S.C. Williams, A general decomposition theory for random cascades, Bull AMS 31 (2) (1994), 216-222, MR 95a:60065.

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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

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