A cascade decomposition theory with applications to Markov and exchangeable cascades
HTML articles powered by AMS MathViewer
- by Edward C. Waymire and Stanley C. Williams PDF
- Trans. Amer. Math. Soc. 348 (1996), 585-632 Request permission
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.References
- Rabi N. Bhattacharya and Edward C. Waymire, Stochastic processes with applications, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1990. A Wiley-Interscience Publication. MR 1054645
- Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
- P. Collet and F. Koukiou, Large deviations for multiplicative chaos, Comm. Math. Phys. 147 (1992), no. 2, 329–342. MR 1174416, DOI 10.1007/BF02096590
- Donald L. Cohn, Measure theory, Birkhäuser, Boston, Mass., 1980. MR 578344, DOI 10.1007/978-1-4899-0399-0
- Colleen D. Cutler, The Hausdorff dimension distribution of finite measures in Euclidean space, Canad. J. Math. 38 (1986), no. 6, 1459–1484. MR 873419, DOI 10.4153/CJM-1986-071-9 Derrida, B., Mean field theory of directed polymers in a random medium and beyond, Physica Scripta T38 (1991), 6-12.
- B. Derrida and H. Spohn, Polymers on disordered trees, spin glasses, and traveling waves, J. Statist. Phys. 51 (1988), no. 5-6, 817–840. New directions in statistical mechanics (Santa Barbara, CA, 1987). MR 971033, DOI 10.1007/BF01014886
- Donald A. Dawson and Edwin A. Perkins, Historical processes, Mem. Amer. Math. Soc. 93 (1991), no. 454, iv+179. MR 1079034, DOI 10.1090/memo/0454
- Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
- Richard Durrett and Thomas M. Liggett, Fixed points of the smoothing transformation, Z. Wahrsch. Verw. Gebiete 64 (1983), no. 3, 275–301. MR 716487, DOI 10.1007/BF00532962
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- E. B. Dynkin, Superdiffusions and parabolic nonlinear differential equations, Ann. Probab. 20 (1992), no. 2, 942–962. MR 1159580, DOI 10.1214/aop/1176989812 Falconer,K., The multifractal spectrum of statistically self- similar measures, preprint (1993). Gupta, V.K. and E. Waymire, A statistical analysis of mesoscale rainfall as a random cascade, Jour. Appld. Meteor. 32 (2) (1993), 251-267.
- Siegfried Graf, R. Daniel Mauldin, and S. C. Williams, The exact Hausdorff dimension in random recursive constructions, Mem. Amer. Math. Soc. 71 (1988), no. 381, x+121. MR 920961, DOI 10.1090/memo/0381
- Yves Guivarc’h, Sur une extension de la notion de loi semi-stable, Ann. Inst. H. Poincaré Probab. Statist. 26 (1990), no. 2, 261–285 (French, with English summary). MR 1063751
- Theodore E. Harris, The theory of branching processes, Die Grundlehren der mathematischen Wissenschaften, Band 119, Springer-Verlag, Berlin; Prentice Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0163361, DOI 10.1007/978-3-642-51866-9
- Richard Holley and Thomas M. Liggett, Generalized potlatch and smoothing processes, Z. Wahrsch. Verw. Gebiete 55 (1981), no. 2, 165–195. MR 608015, DOI 10.1007/BF00535158
- Richard Holley and Edward C. Waymire, Multifractal dimensions and scaling exponents for strongly bounded random cascades, Ann. Appl. Probab. 2 (1992), no. 4, 819–845. MR 1189419
- 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
- Jean-Pierre Kahane, Multiplications aléatoires et dimensions de Hausdorff, Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), no. 2, suppl., 289–296 (French, with English summary). MR 898497
- Jean-Pierre Kahane, Positive martingales and random measures, Chinese Ann. Math. Ser. B 8 (1987), no. 1, 1–12. A Chinese summary appears in Chinese Ann. Math. Ser. A 8 (1987), no. 1, 136. MR 886744
- Jean-Pierre Kahane, From Riesz products to random sets, Harmonic analysis (Sendai, 1990) ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991, pp. 125–139. MR 1261434
- 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
- Jean-Pierre Kahane and Yitzhak Katznelson, Décomposition des mesures selon la dimension, Colloq. Math. 58 (1990), no. 2, 269–279 (French). MR 1060178, DOI 10.4064/cm-58-2-269-279
- Harry Kesten, Subdiffusive behavior of random walk on a random cluster, Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 4, 425–487 (English, with French summary). MR 871905
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089, DOI 10.1007/BF03046994 Koukiou, F., The mean-field theory of spin glass and directed polymer models in random media, preprint (1993).
- Torgny Lindvall, Lectures on the coupling method, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1992. A Wiley-Interscience Publication. MR 1180522
- Russell Lyons, Random walks and percolation on trees, Ann. Probab. 18 (1990), no. 3, 931–958. MR 1062053
- R. Daniel Mauldin and S. C. Williams, Random recursive constructions: asymptotic geometric and topological properties, Trans. Amer. Math. Soc. 295 (1986), no. 1, 325–346. MR 831202, DOI 10.1090/S0002-9947-1986-0831202-5 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.
- Steven Orey, Lecture notes on limit theorems for Markov chain transition probabilities, Van Nostrand Reinhold Mathematical Studies, No. 34, Van Nostrand Reinhold Co., London-New York-Toronto, Ont., 1971. MR 0324774
- Jacques Peyrière, Calculs de dimensions de Hausdorff, Duke Math. J. 44 (1977), no. 3, 591–601. MR 444911
- L. A. Shepp, Covering the line with random intervals, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 23 (1972), 163–170. MR 322923, DOI 10.1007/BF00536556
- D. Lavallée, S. Lovejoy, D. Schertzer, and F. Schmitt, On the determination of universal multifractal parameters in turbulence, Topological aspects of the dynamics of fluids and plasmas (Santa Barbara, CA, 1991) NATO Adv. Sci. Inst. Ser. E: Appl. Sci., vol. 218, Kluwer Acad. Publ., Dordrecht, 1992, pp. 463–478. MR 1232248, DOI 10.1007/978-94-017-3550-6_{2}7 Waymire, E. and S.C. Williams, Markov cascades, IMA Volume on Branching Processes, ed by K. Athreya and P. Jagers, in press., 1995. Waymire, E. and S.C. Williams, Multiplicative cascades: dimension spectra and dependence, special issue, Journal of Fourier Analysis and Applications (1995), 589-609. Waymire, E. and S.C. Williams, Correlated spin glasses on trees, preprint (1995).
- Edward C. Waymire and Stanley C. Williams, A general decomposition theory for random cascades, Bull. Amer. Math. Soc. (N.S.) 31 (1994), no. 2, 216–222. MR 1260522, DOI 10.1090/S0273-0979-1994-00521-X
Additional Information
- Edward C. Waymire
- MR Author ID: 180975
- Email: waymire@math.orst.edu
- Stanley C. Williams
- Email: williams@sunfs.math.usu.edu
- 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
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 585-632
- MSC (1991): Primary 60G57, 60G30, 60G42; Secondary 60K35, 60D05, 60J10, 60G09
- DOI: https://doi.org/10.1090/S0002-9947-96-01500-0
- MathSciNet review: 1322959