|
A general decomposition theory for random cascades
Author(s):
Edward C.
Waymire;
Stanley C.
Williams
Journal:
Bull. Amer. Math. Soc.
31
(1994),
216-222.
MathSciNet review:
1260522
Retrieve article in:
PDF
Abstract |
References |
Additional information
Abstract:
This announcement describes a probabilistic approach to cascades which, in addition to providing an entirely probabilistic proof of the Kahane-Peyrière theorem for independent cascades, readily applies to general dependent cascades. Moreover, this unifies various seemingly disparate cascade decompositions, including Kahane's T-martingale decomposition and dimension disintegration.
References:
-
- [C]
- C. Cutler, The Hausdorff dimension distribution of finite measures in Euclidean space, Canad. J. Math. XXXVIII (1986). MR 873419 (88b:28013)
- [CK]
- P. Collet and F. Koukiou, Large deviations for multiplicative chaos, Comm. Math. Phys. 147 (1992), 329-342. MR 1174416 (94a:82003)
- [CCD]
- J. Chayes, L. Chayes, and R. Durrett, Connectivity properties of Mandelbrot's percolation, Probab. Theory Related Fields 77 (1988), 307-324. MR 931500 (89d:60193)
- [DE]
- B. Derrida, Directed polymers in a random medium, Phys. A 163 (1990), 71-84. MR 1043640 (90m:82038)
- [DF]
- L. Dubins and D. A. Freedman, Random distribution functions, Bull. Amer. Math. Soc. 69 (1963), 548-551. MR 0149522 (26:7008)
- [DG]
- F. M. Dekking and G. R. Grimmett, Superbranching processes and projections of random Cantor sets, Probab. Theory Related Fields 78 (1988), 335-355. MR 949177 (89f:60099)
- [DL]
- R. Durrett and T. M. Liggett, Fixed points of the smoothing transformation, Z. Wahr. Verw. Geb. 64 (1983), 275-301. MR 716487 (85e:60059)
- [DM]
- F. M. Dekking and R. W. J. Meester, On the structure of Mandelbrot's percolation process and other random Cantor sets, J. Stat. Phys. 58 (1990), 1109-1126. MR 1049059 (91c:60140)
- [F]
- K. Falconer, The multifractal spectrum of statistically self-similar measures, preprint. MR 1284660 (95m:60076)
- [GMW]
- S. Graf, D. Mauldin, and S. Williams, The exact Hausdorff dimension in random recursive constructions, Mem. Amer. Math. Soc., vol. 71, Amer. Math. Soc., Providence, RI, 1987. MR 920961 (88k:28010)
- [GW]
- V. K. Gupta and E. Waymire, A statistical analysis of mesoscale rainfall as a random cascade, J. Appl. Meteor. 32 (1993), 251-267.
- [HL]
- R. Holley and T. M. Liggett, Generalized potlatch and smoothing processes, Z. Wahr Verw. Geb. 55 (1981), 165-195. MR 608015 (82i:60176)
- [HW]
- R. Holley and E. Waymire, Multifractal dimensions and scaling exponents for strongly bounded random cascades, Ann. Appl. Probab. 2 (1992), 819-845. MR 1189419 (93k:60122)
- [KP]
- J. P. Kahane and J. Peyrière, Sur certaines martingales de Benoît Mandelbrot, Adv. Math. 22 (1976), 131-145. MR 0431355 (55:4355)
- [K1]
- J. P. Kahane, Multiplications aléatoires et dimensions de Hausdorff, Ann. Inst. Poincaré 23 (1987), 289-296. MR 898497 (88h:60100)
- [K2]
- -, Positive Martingales and random measures, Chinese Ann. Math. 8 (1987), 1-12. MR 886744 (88j:60098)
- [K3]
- -, Random multiplications, random coverings, and multiplicative chaos, Proceedings of the Special Year in Modern Analysis (E. Berkson, N. Tenney Peck, J. Jerry Uhl, eds.), London Math. Soc. Lect. Notes, vol. 137, Cambridge Univ. Press, London, 1989, pp. 196-255. MR 1009176 (91e:60152)
- [K4]
- -, Produits de poids aléatoires indépendants et applications, Fractal Geometry and Analysis (J. Belair and S. Dubac, eds.), Kluwer Academic Publ., The Netherlands, 1991. MR 1140725
- [KK]
- J. P. Kahane and Y. Katznelson, Décomposition des mesures selon la dimension, Colloq. Math. LVII (1990), 269-279. MR 1060178 (91g:28018)
- [L]
- R. Lyons, Random walks and percolation on trees, Ann. Probab. 18 (1990), 931-958. MR 1062053 (91i:60179)
- [MS]
- C. Meneveau and K. R. Sreenivasan, The multifractal spectrum of the dissipation field in turbulent flows, Nuclear Phys. B (Proc. Suppl.) 2 (1987), 49-76.
- [N]
- Ben Nasr, Mesures aléatores de Mandelbrot associées à des substitutions, C. R. Acad. Sc. Paris Sér. I Math. 304 (1987), 255-258. MR 882783 (88b:60124)
- [O]
- L. Olsen, Random geometrically graph directed self-similar multifractals, Mem. Amer. Math. Soc. (to appear). MR 1297123 (95j:28006)
- [PW]
- S. Pekham and E. Waymire, On a symmetry of turbulence, Comm. Math. Phys 147 (1991), 365-370. MR 1174418 (93c:76053)
- [P]
- J. Peyrière, Calculs de dimensions de Hausdorff, Duke Math. J. 44 (1977), 591-601. MR 0444911 (56:3257)
- [Pr]
- C. Preston, Gibbs states on countable sets, Cambridge Univ. Press, London, 1974. MR 0474556 (57:14194a)
- [S]
- L. Shepp, Covering the line with random intervals, Z. Wahr. Verw. Geb. 23 (1972), 163-170. MR 0322923 (48:1284)
- [Sp]
- F. Spitzer, Markov random fields on an infinite tree, Ann. Probab. 3 (1975), 387-398. MR 0378152 (51:14321)
- [TLS]
- Y. Tessier, S. Lovejoy, and D. Schertzer, Universal multifractals: Theory and observation for rain and clouds, J. Appl. Meteor. 32 (1993), 223-250.
- [WW1]
- E. Waymire and S. C. Williams, Markov cascades, preprint. MR 1601698 (99k:60093)
- [WW2]
- -, Multiplicative cascades : Dimension spectra and dependence, Preprint presented at Colloque en l'honneur de J.-P. Kahane (June 28 - July 3, 1993), Université de Paris-Sud, Orsay, France.
- [WW3]
- -, A cascade decomposition theory with applications to Markov and exchangeable cascades, preprint.
Additional Information:
DOI:
10.1090/S0273-0979-1994-00521-X
PII:
S 0273-0979(1994)00521-X
Keywords:
Martingale,
Hausdorff dimension,
tree,
random measure
Copyright of article:
Copyright
1994,
American Mathematical Society
|
