DOI: 10.1090/s0273-0979-1994-00521-x. A general decomposition theory for random cascades

A general decomposition theory for random cascades

10.1090/s0273-0979-1994-00521-x

Crossref journal-article

American Mathematical Society (AMS)

Bulletin of the American Mathematical Society (14)

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.

Bibliography

Waymire, E. C., & Williams, S. C. (1994). A general decomposition theory for random cascades. Bulletin of the American Mathematical Society, 31(2), 216â222. Portico.

Authors 2

Edward C. Waymire (first)
Stanley C. Williams (additional)

References 32 Referenced 26

10.4153/CJM-1986-071-9 / Canad. J. Math. / The Hausdorff dimension distribution of finite measures in Euclidean space by Cutler, Colleen D. (1986)
10.1007/BF02096590 / Comm. Math. Phys. / Large deviations for multiplicative chaos by Collet, P. (1992)
10.1007/BF00319291 / Probab. Theory Related Fields / Connectivity properties of Mandelbrot’s percolation process by Chayes, J. T. (1988)
10.1016/0378-4371(90)90316-K / Phys. A / Directed polymers in a random medium by Derrida, B. (1990)
10.1090/S0002-9904-1963-10992-1 / Bull. Amer. Math. Soc. / Random distribution functions by Dubins, Lester E. (1963)
10.1007/BF00334199 / Probab. Theory Related Fields / Superbranching processes and projections of random Cantor sets by Dekking, F. M. (1988)
10.1007/BF00532962 / Z. Wahrsch. Verw. Gebiete / Fixed points of the smoothing transformation by Durrett, Richard (1983)
10.1007/BF01026566 / J. Statist. Phys. / On the structure of Mandelbrot’s percolation process and other random Cantor sets by Dekking, F. M. (1990)
10.1007/BF02213576 / J. Theoret. Probab. / The multifractal spectrum of statistically self-similar measures by Falconer, K. J. (1994)
10.1090/memo/0381 / Mem. Amer. Math. Soc. / The exact Hausdorff dimension in random recursive constructions by Graf, Siegfried (1988)
V. K. Gupta and E. Waymire, A statistical analysis of mesoscale rainfall as a random cascade, J. Appl. Meteor. 32 (1993), 251-267. (10.1175/1520-0450(1993)032<0251:ASAOMR>2.0.CO;2)
10.1007/BF00535158 / Z. Wahrsch. Verw. Gebiete / Generalized potlatch and smoothing processes by Holley, Richard (1981)
{'issue': '4', 'key': '13', 'first-page': '819', 'article-title': 'Multifractal dimensions and scaling exponents for strongly bounded random cascades', 'volume': '2', 'author': 'Holley, Richard', 'year': '1992', 'journal-title': 'Ann. Appl. Probab.', 'ISSN': 'http://id.crossref.org/issn/1050-5164', 'issn-type': 'print'} / Ann. Appl. Probab. / Multifractal dimensions and scaling exponents for strongly bounded random cascades by Holley, Richard (1992)
10.1016/0001-8708(76)90151-1 / Advances in Math. / Sur certaines martingales de Benoit Mandelbrot by Kahane, J.-P. (1976)
{'issue': '2', 'key': '15', 'first-page': '289', 'article-title': 'Multiplications aléatoires et dimensions de Hausdorff', 'volume': '23', 'author': 'Kahane, Jean-Pierre', 'year': '1987', 'journal-title': "Ann. Inst. H. Poincar\\'{e} Probab. Statist.", 'ISSN': 'http://id.crossref.org/issn/0246-0203', 'issn-type': 'print'} / Ann. Inst. H. Poincar\'{e} Probab. Statist. / Multiplications aléatoires et dimensions de Hausdorff by Kahane, Jean-Pierre (1987)
{'issue': '1', 'key': '16', 'first-page': '1', 'article-title': 'Positive martingales and random measures', 'volume': '8', 'author': 'Kahane, Jean-Pierre', 'year': '1987', 'journal-title': 'Chinese Ann. Math. Ser. B', 'ISSN': 'http://id.crossref.org/issn/0252-9599', 'issn-type': 'print'} / Chinese Ann. Math. Ser. B / Positive martingales and random measures by Kahane, Jean-Pierre (1987)
{'key': '17', 'first-page': '196', 'article-title': 'Random multiplications, random coverings, multiplicative chaos', 'author': 'Kahane, Jean-Pierre', 'year': '1989'} / Random multiplications, random coverings, multiplicative chaos by Kahane, Jean-Pierre (1989)
{'key': '18', 'first-page': '277', 'article-title': 'Produits de poids aléatoires indépendants et applications', 'author': 'Kahane, Jean-Pierre', 'year': '1991'} / Produits de poids aléatoires indépendants et applications by Kahane, Jean-Pierre (1991)
10.4064/cm-58-2-269-279 / Colloq. Math. / Décomposition des mesures selon la dimension by Kahane, Jean-Pierre (1990)
{'issue': '3', 'key': '20', 'first-page': '931', 'article-title': 'Random walks and percolation on trees', 'volume': '18', 'author': 'Lyons, Russell', 'year': '1990', 'journal-title': 'Ann. Probab.', 'ISSN': 'http://id.crossref.org/issn/0091-1798', 'issn-type': 'print'} / Ann. Probab. / Random walks and percolation on trees by Lyons, Russell (1990)
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. (10.1016/0920-5632(87)90008-9)
{'issue': '10', 'key': '22', 'first-page': '255', 'article-title': 'Mesures aléatoires de Mandelbrot associées à des substitutions', 'volume': '304', 'author': 'Ben Nasr, Fathi', 'year': '1987', 'journal-title': "C. R. Acad. Sci. Paris S\\'{e}r. I Math.", 'ISSN': 'http://id.crossref.org/issn/0249-6291', 'issn-type': 'print'} / C. R. Acad. Sci. Paris S\'{e}r. I Math. / Mesures aléatoires de Mandelbrot associées à des substitutions by Ben Nasr, Fathi (1987)
10.2307/4351476 / Random geometrically graph directed self-similar multifractals / Pitman Research Notes in Mathematics Series by Olsen, Lars (1994)
10.1007/BF02096592 / Comm. Math. Phys. / On a symmetry of turbulence by Peckham, Scott (1992)
{'issue': '3', 'key': '25', 'first-page': '591', 'article-title': 'Calculs de dimensions de Hausdorff', 'volume': '44', 'author': 'Peyrière, Jacques', 'year': '1977', 'journal-title': 'Duke Math. J.', 'ISSN': 'http://id.crossref.org/issn/0012-7094', 'issn-type': 'print'} / Duke Math. J. / Calculs de dimensions de Hausdorff by Peyrière, Jacques (1977)
10.1017/CBO9780511897122 / Gibbs states on countable sets / Cambridge Tracts in Mathematics, No. 68 by Preston, Christopher J. (1974)
10.1007/BF00536556 / Z. Wahrscheinlichkeitstheorie und Verw. Gebiete / Covering the line with random intervals by Shepp, L. A. (1972)
10.1214/aop/1176996347 / Ann. Probability / Markov random fields on an infinite tree by Spitzer, Frank (1975)
Y. Tessier, S. Lovejoy, and D. Schertzer, Universal multifractals: Theory and observation for rain and clouds, J. Appl. Meteor. 32 (1993), 223-250. (10.1175/1520-0450(1993)032<0223:UMTAOF>2.0.CO;2)
10.1007/978-1-4612-1862-3_24 / Markov cascades by Waymire, Edward C. (1997)
\bysame, 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.
\bysame, A cascade decomposition theory with applications to Markov and exchangeable cascades, preprint.

Dates

Type	When
Created	17 years, 7 months ago (Jan. 28, 2008, 8:50 a.m.)
Deposited	3 years, 10 months ago (Oct. 26, 2021, 9:41 a.m.)
Indexed	1 year, 8 months ago (Dec. 12, 2023, 2:23 a.m.)
Issued	31 years, 8 months ago (Jan. 1, 1994)
Published	31 years, 8 months ago (Jan. 1, 1994)
Published Online	31 years, 8 months ago (Jan. 1, 1994)

Funders 0

None

JSON

{
  "indexed": {
    "date-parts": [
      [
        2023,
        12,
        12
      ]
    ],
    "date-time": "2023-12-12T07:23:25Z",
    "timestamp": 1702365805209
  },
  "reference-count": 32,
  "publisher": "American Mathematical Society (AMS)",
  "issue": "2",
  "content-domain": {
    "domain": [],
    "crossmark-restriction": false
  },
  "abstract": "<p>This announcement describes a probabilistic approach to cascades which, in addition to providing an entirely probabilistic proof of the Kahane-Peyri\u00e8re theorem for independent cascades, readily applies to general dependent cascades. Moreover, this unifies various seemingly disparate cascade decompositions, including Kahane\u2019s T-martingale decomposition and dimension disintegration.</p>",
  "DOI": "10.1090/s0273-0979-1994-00521-x",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2008,
        1,
        28
      ]
    ],
    "date-time": "2008-01-28T13:50:56Z",
    "timestamp": 1201528256000
  },
  "page": "216-222",
  "source": "Crossref",
  "is-referenced-by-count": 26,
  "title": "A general decomposition theory for random cascades",
  "prefix": "10.1090",
  "volume": "31",
  "author": [
    {
      "given": "Edward C.",
      "family": "Waymire",
      "sequence": "first",
      "affiliation": []
    },
    {
      "given": "Stanley C.",
      "family": "Williams",
      "sequence": "additional",
      "affiliation": []
    }
  ],
  "member": "14",
  "published-online": {
    "date-parts": [
      [
        1994
      ]
    ]
  },
  "reference": [
    {
      "issue": "6",
      "key": "1",
      "doi-asserted-by": "publisher",
      "first-page": "1459",
      "DOI": "10.4153/CJM-1986-071-9",
      "article-title": "The Hausdorff dimension distribution of finite measures in Euclidean space",
      "volume": "38",
      "author": "Cutler, Colleen D.",
      "year": "1986",
      "journal-title": "Canad. J. Math.",
      "ISSN": "http://id.crossref.org/issn/0008-414X",
      "issn-type": "print"
    },
    {
      "issue": "2",
      "key": "2",
      "doi-asserted-by": "crossref",
      "first-page": "329",
      "DOI": "10.1007/BF02096590",
      "article-title": "Large deviations for multiplicative chaos",
      "volume": "147",
      "author": "Collet, P.",
      "year": "1992",
      "journal-title": "Comm. Math. Phys.",
      "ISSN": "http://id.crossref.org/issn/0010-3616",
      "issn-type": "print"
    },
    {
      "issue": "3",
      "key": "3",
      "doi-asserted-by": "publisher",
      "first-page": "307",
      "DOI": "10.1007/BF00319291",
      "article-title": "Connectivity properties of Mandelbrot\u2019s percolation process",
      "volume": "77",
      "author": "Chayes, J. T.",
      "year": "1988",
      "journal-title": "Probab. Theory Related Fields",
      "ISSN": "http://id.crossref.org/issn/0178-8051",
      "issn-type": "print"
    },
    {
      "issue": "1",
      "key": "4",
      "doi-asserted-by": "publisher",
      "first-page": "71",
      "DOI": "10.1016/0378-4371(90)90316-K",
      "article-title": "Directed polymers in a random medium",
      "volume": "163",
      "author": "Derrida, B.",
      "year": "1990",
      "journal-title": "Phys. A",
      "ISSN": "http://id.crossref.org/issn/0378-4371",
      "issn-type": "print"
    },
    {
      "key": "5",
      "doi-asserted-by": "publisher",
      "first-page": "548",
      "DOI": "10.1090/S0002-9904-1963-10992-1",
      "article-title": "Random distribution functions",
      "volume": "69",
      "author": "Dubins, Lester E.",
      "year": "1963",
      "journal-title": "Bull. Amer. Math. Soc.",
      "ISSN": "http://id.crossref.org/issn/0002-9904",
      "issn-type": "print"
    },
    {
      "issue": "3",
      "key": "6",
      "doi-asserted-by": "publisher",
      "first-page": "335",
      "DOI": "10.1007/BF00334199",
      "article-title": "Superbranching processes and projections of random Cantor sets",
      "volume": "78",
      "author": "Dekking, F. M.",
      "year": "1988",
      "journal-title": "Probab. Theory Related Fields",
      "ISSN": "http://id.crossref.org/issn/0178-8051",
      "issn-type": "print"
    },
    {
      "issue": "3",
      "key": "7",
      "doi-asserted-by": "publisher",
      "first-page": "275",
      "DOI": "10.1007/BF00532962",
      "article-title": "Fixed points of the smoothing transformation",
      "volume": "64",
      "author": "Durrett, Richard",
      "year": "1983",
      "journal-title": "Z. Wahrsch. Verw. Gebiete",
      "ISSN": "http://id.crossref.org/issn/0044-3719",
      "issn-type": "print"
    },
    {
      "issue": "5-6",
      "key": "8",
      "doi-asserted-by": "publisher",
      "first-page": "1109",
      "DOI": "10.1007/BF01026566",
      "article-title": "On the structure of Mandelbrot\u2019s percolation process and other random Cantor sets",
      "volume": "58",
      "author": "Dekking, F. M.",
      "year": "1990",
      "journal-title": "J. Statist. Phys.",
      "ISSN": "http://id.crossref.org/issn/0022-4715",
      "issn-type": "print"
    },
    {
      "issue": "3",
      "key": "9",
      "doi-asserted-by": "publisher",
      "first-page": "681",
      "DOI": "10.1007/BF02213576",
      "article-title": "The multifractal spectrum of statistically self-similar measures",
      "volume": "7",
      "author": "Falconer, K. J.",
      "year": "1994",
      "journal-title": "J. Theoret. Probab.",
      "ISSN": "http://id.crossref.org/issn/0894-9840",
      "issn-type": "print"
    },
    {
      "issue": "381",
      "key": "10",
      "doi-asserted-by": "publisher",
      "first-page": "x+121",
      "DOI": "10.1090/memo/0381",
      "article-title": "The exact Hausdorff dimension in random recursive constructions",
      "volume": "71",
      "author": "Graf, Siegfried",
      "year": "1988",
      "journal-title": "Mem. Amer. Math. Soc.",
      "ISSN": "http://id.crossref.org/issn/0065-9266",
      "issn-type": "print"
    },
    {
      "key": "11",
      "doi-asserted-by": "crossref",
      "unstructured": "V. K. Gupta and E. Waymire, A statistical analysis of mesoscale rainfall as a random cascade, J. Appl. Meteor. 32 (1993), 251-267.",
      "DOI": "10.1175/1520-0450(1993)032<0251:ASAOMR>2.0.CO;2"
    },
    {
      "issue": "2",
      "key": "12",
      "doi-asserted-by": "publisher",
      "first-page": "165",
      "DOI": "10.1007/BF00535158",
      "article-title": "Generalized potlatch and smoothing processes",
      "volume": "55",
      "author": "Holley, Richard",
      "year": "1981",
      "journal-title": "Z. Wahrsch. Verw. Gebiete",
      "ISSN": "http://id.crossref.org/issn/0044-3719",
      "issn-type": "print"
    },
    {
      "issue": "4",
      "key": "13",
      "first-page": "819",
      "article-title": "Multifractal dimensions and scaling exponents for strongly bounded random cascades",
      "volume": "2",
      "author": "Holley, Richard",
      "year": "1992",
      "journal-title": "Ann. Appl. Probab.",
      "ISSN": "http://id.crossref.org/issn/1050-5164",
      "issn-type": "print"
    },
    {
      "issue": "2",
      "key": "14",
      "doi-asserted-by": "publisher",
      "first-page": "131",
      "DOI": "10.1016/0001-8708(76)90151-1",
      "article-title": "Sur certaines martingales de Benoit Mandelbrot",
      "volume": "22",
      "author": "Kahane, J.-P.",
      "year": "1976",
      "journal-title": "Advances in Math.",
      "ISSN": "http://id.crossref.org/issn/0001-8708",
      "issn-type": "print"
    },
    {
      "issue": "2",
      "key": "15",
      "first-page": "289",
      "article-title": "Multiplications al\u00e9atoires et dimensions de Hausdorff",
      "volume": "23",
      "author": "Kahane, Jean-Pierre",
      "year": "1987",
      "journal-title": "Ann. Inst. H. Poincar\\'{e} Probab. Statist.",
      "ISSN": "http://id.crossref.org/issn/0246-0203",
      "issn-type": "print"
    },
    {
      "issue": "1",
      "key": "16",
      "first-page": "1",
      "article-title": "Positive martingales and random measures",
      "volume": "8",
      "author": "Kahane, Jean-Pierre",
      "year": "1987",
      "journal-title": "Chinese Ann. Math. Ser. B",
      "ISSN": "http://id.crossref.org/issn/0252-9599",
      "issn-type": "print"
    },
    {
      "key": "17",
      "first-page": "196",
      "article-title": "Random multiplications, random coverings, multiplicative chaos",
      "author": "Kahane, Jean-Pierre",
      "year": "1989"
    },
    {
      "key": "18",
      "first-page": "277",
      "article-title": "Produits de poids al\u00e9atoires ind\u00e9pendants et applications",
      "author": "Kahane, Jean-Pierre",
      "year": "1991"
    },
    {
      "issue": "2",
      "key": "19",
      "doi-asserted-by": "publisher",
      "first-page": "269",
      "DOI": "10.4064/cm-58-2-269-279",
      "article-title": "D\u00e9composition des mesures selon la dimension",
      "volume": "58",
      "author": "Kahane, Jean-Pierre",
      "year": "1990",
      "journal-title": "Colloq. Math.",
      "ISSN": "http://id.crossref.org/issn/0010-1354",
      "issn-type": "print"
    },
    {
      "issue": "3",
      "key": "20",
      "first-page": "931",
      "article-title": "Random walks and percolation on trees",
      "volume": "18",
      "author": "Lyons, Russell",
      "year": "1990",
      "journal-title": "Ann. Probab.",
      "ISSN": "http://id.crossref.org/issn/0091-1798",
      "issn-type": "print"
    },
    {
      "key": "21",
      "doi-asserted-by": "crossref",
      "unstructured": "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.",
      "DOI": "10.1016/0920-5632(87)90008-9"
    },
    {
      "issue": "10",
      "key": "22",
      "first-page": "255",
      "article-title": "Mesures al\u00e9atoires de Mandelbrot associ\u00e9es \u00e0 des substitutions",
      "volume": "304",
      "author": "Ben Nasr, Fathi",
      "year": "1987",
      "journal-title": "C. R. Acad. Sci. Paris S\\'{e}r. I Math.",
      "ISSN": "http://id.crossref.org/issn/0249-6291",
      "issn-type": "print"
    },
    {
      "key": "23",
      "series-title": "Pitman Research Notes in Mathematics Series",
      "isbn-type": "print",
      "doi-asserted-by": "publisher",
      "DOI": "10.2307/4351476",
      "volume-title": "Random geometrically graph directed self-similar multifractals",
      "volume": "307",
      "author": "Olsen, Lars",
      "year": "1994",
      "ISBN": "http://id.crossref.org/isbn/0582253810"
    },
    {
      "issue": "2",
      "key": "24",
      "doi-asserted-by": "crossref",
      "first-page": "365",
      "DOI": "10.1007/BF02096592",
      "article-title": "On a symmetry of turbulence",
      "volume": "147",
      "author": "Peckham, Scott",
      "year": "1992",
      "journal-title": "Comm. Math. Phys.",
      "ISSN": "http://id.crossref.org/issn/0010-3616",
      "issn-type": "print"
    },
    {
      "issue": "3",
      "key": "25",
      "first-page": "591",
      "article-title": "Calculs de dimensions de Hausdorff",
      "volume": "44",
      "author": "Peyri\u00e8re, Jacques",
      "year": "1977",
      "journal-title": "Duke Math. J.",
      "ISSN": "http://id.crossref.org/issn/0012-7094",
      "issn-type": "print"
    },
    {
      "key": "26",
      "series-title": "Cambridge Tracts in Mathematics, No. 68",
      "doi-asserted-by": "crossref",
      "DOI": "10.1017/CBO9780511897122",
      "volume-title": "Gibbs states on countable sets",
      "author": "Preston, Christopher J.",
      "year": "1974"
    },
    {
      "key": "27",
      "doi-asserted-by": "publisher",
      "first-page": "163",
      "DOI": "10.1007/BF00536556",
      "article-title": "Covering the line with random intervals",
      "volume": "23",
      "author": "Shepp, L. A.",
      "year": "1972",
      "journal-title": "Z. Wahrscheinlichkeitstheorie und Verw. Gebiete"
    },
    {
      "issue": "3",
      "key": "28",
      "doi-asserted-by": "publisher",
      "first-page": "387",
      "DOI": "10.1214/aop/1176996347",
      "article-title": "Markov random fields on an infinite tree",
      "volume": "3",
      "author": "Spitzer, Frank",
      "year": "1975",
      "journal-title": "Ann. Probability",
      "ISSN": "http://id.crossref.org/issn/0091-1798",
      "issn-type": "print"
    },
    {
      "key": "29",
      "doi-asserted-by": "crossref",
      "unstructured": "Y. Tessier, S. Lovejoy, and D. Schertzer, Universal multifractals: Theory and observation for rain and clouds, J. Appl. Meteor. 32 (1993), 223-250.",
      "DOI": "10.1175/1520-0450(1993)032<0223:UMTAOF>2.0.CO;2"
    },
    {
      "key": "30",
      "doi-asserted-by": "publisher",
      "first-page": "305",
      "DOI": "10.1007/978-1-4612-1862-3_24",
      "article-title": "Markov cascades",
      "author": "Waymire, Edward C.",
      "year": "1997"
    },
    {
      "key": "31",
      "unstructured": "\\bysame, Multiplicative cascades : Dimension spectra and dependence, Preprint presented at Colloque en l\u2019honneur de J.-P. Kahane (June 28 - July 3, 1993), Universit\u00e9 de Paris-Sud, Orsay, France."
    },
    {
      "key": "32",
      "unstructured": "\\bysame, A cascade decomposition theory with applications to Markov and exchangeable cascades, preprint."
    }
  ],
  "container-title": "Bulletin of the American Mathematical Society",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "http://www.ams.org/bull/1994-31-02/S0273-0979-1994-00521-X/S0273-0979-1994-00521-X.pdf",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    },
    {
      "URL": "https://www.ams.org/bull/1994-31-02/S0273-0979-1994-00521-X/S0273-0979-1994-00521-X.pdf",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2021,
        10,
        26
      ]
    ],
    "date-time": "2021-10-26T13:41:31Z",
    "timestamp": 1635255691000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://www.ams.org/bull/1994-31-02/S0273-0979-1994-00521-X/"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        1994
      ]
    ]
  },
  "references-count": 32,
  "journal-issue": {
    "issue": "2",
    "published-print": {
      "date-parts": [
        [
          1994,
          10
        ]
      ]
    }
  },
  "alternative-id": [
    "S0273-0979-1994-00521-X"
  ],
  "URL": "http://dx.doi.org/10.1090/S0273-0979-1994-00521-X",
  "relation": {},
  "ISSN": [
    "0273-0979",
    "1088-9485"
  ],
  "subject": [],
  "container-title-short": "Bull. Amer. Math. Soc.",
  "published": {
    "date-parts": [
      [
        1994
      ]
    ]
  }
}