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Study on semi-Levy processes and semi-selfsimilar processes

Research Project

Project/Area Number 15340036
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKEIO UNIVERSITY

Principal Investigator

MAEJIMA Makoto  Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (90051846)

Co-Investigator(Kenkyū-buntansha) NAKADA Hitoshi  Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (40118980)
TANI Atusi  Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (90118969)
TAMURA Yozo  Keio Univ., Faculty of Sci.and Tech., Associate Professor, 理工学部, 助教授 (50171905)
SATO Ken-iti  Nagoya University, Emeritus, Professor, 名誉教授 (60015500)
WATANABE Toshiro  The University of Aizu, Center of Mathematical Sciences, Lecturer, 総合数理科学センター, 専任講師 (50254115)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥5,900,000 (Direct Cost: ¥5,900,000)
Fiscal Year 2005: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2004: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2003: ¥2,100,000 (Direct Cost: ¥2,100,000)
KeywordsLevy process / semi-Levy process / infinitely divisible distribution / stochastic integral characterization / semi-selfsimilar process / additive parocess / 確立積分表現 / 半定常過程 / 半自己分解可能分布 / オルンシュタイン・ウーレンベック過程
Research Abstract

This year was the last year of the three year project, and we spent most time for the study of stochastic integral characterizations of several subclasses of infinitely divisible disitirutions in connection with the purpose of the whole project. As a result, we succeeded in stochastic integral characterization of the class of type G distributions and its decreasing sequence of subclasses, which was one of the aims at the beginning of the academic year of 2005. Furthermore, we found a new subclass of type G and seldecomposable distributions which has very simple form of Levy measures in the Levy-Khintchine representation of the characteristic function of infinitely divisible distribution. Maejima gave a talk on this result together with more other related results at the seminar of Munich Technical University, Germany, November 2005, and the talk was very much appreciated. Although we were not able to get a big success on semi-Levy processes, Maejima suggested in the seminar the idea of … More replacing Levy process in the stochastic integral by semi-Levy process, and this idea was well accepted by the audience, and Maejima felt a wave of a new international joint research on this topic. These works were mainly done by Maejima and Sato, Maejima, his student and Jan Rosinski at University of Tenessee.
Another topic which was mentioned at the beginning of this academic year was a new topic, which is generalized Ornstein-Uhlenbeck processes. We investigated the infinite divisibility of the limiting distributions of generalized Ornstein-Uhlenbeck processes and we found that under a mild condition they are not only infinitely divisible but also seldecomposable. Also several interesting examples were constructed. These results were presented by Maejima at an International Conference in Mexico, March 2006.
Based on the results obtained during the three year project, we can see the route of the study of Semi-Levy processes and more precise classification of the class of infinitely divisible distributions. Besides those, Maejima found, together with Victor Perez-Abrue at Maxico, examples of random matrices whose determinants are infinitely divisible. These examples are interesting because it is known that the original random matrices are not infinitely divisible. A lot of interesting unsolved problems around the infinitely divisibility remain. Less

Report

(4 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (26 results)

All 2006 2005 2004 Other

All Journal Article (20 results) Publications (6 results)

  • [Journal Article] Some classes of infinitely divisible distributions admitting stochastic integral representations2006

    • Author(s)
      O.E.Barndorff-Nielsen, M.Taejima, K.Sato
    • Journal Title

      Bernoulli Vol.12, No.1

      Pages: 1-33

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Operator semi-self-similar processes and their space-scaling matrices2006

    • Author(s)
      T.Saigo, Y.Tamura
    • Journal Title

      Statistics and Probability Letters Vol.76, No.7

      Pages: 675-681

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Someclasses of infinitely divisible distributions admitting stochastic integral representations2006

    • Author(s)
      O.E.Barndorff-Nielsen, M.Maejima, K.Sato
    • Journal Title

      Bernoulli Vol.12, No.1

      Pages: 1-33

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Some classes of infinitely divisible distributions admitting stochastic integral representations2006

    • Author(s)
      O.E.Barndorff-Nielsen, M.Maejima, K.Sato
    • Journal Title

      Bernoulli Vol.12, No.1

      Pages: 1-33

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Infinite divisibility and generalized subexponentiality2005

    • Author(s)
      T.Shimura, T.Watanabe
    • Journal Title

      Bernoulli Vol.11, No.3

      Pages: 445-469

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Exact packing measure on the boundary of a Galton-Watson tree2004

    • Author(s)
      T.Watanabe
    • Journal Title

      J.London Math.Soc. vol.69

      Pages: 801-816

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Infinite divisibility for stochastic processes and time change

    • Author(s)
      O.E..Barndorff-Nielsen, M.Maejima, K.Sato
    • Journal Title

      J.Theoretical Probability to appear

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Moments and projections of semistable probability measures on p-adic vector spaces

    • Author(s)
      M.Maejima, R.Shah
    • Journal Title

      J.Theoretical Probability to appear

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Two families of improper stochastic integrals with respect to Levy processes

    • Author(s)
      K.Sato
    • Journal Title

      Latin American Journal of Probability and Matheamtical Statistics to appear

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Infinite divisibility for stochastic processes and time change

    • Author(s)
      O.E.Barndorff-Nielsen, M.Maejima, K.Sato
    • Journal Title

      J.Theoretical Probability (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Moments and projections of semistable probability measures on p-adic vector spaces

    • Author(s)
      M.Maejima, R.Shah
    • Journal Title

      J.Theoretical Probability (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Two families of improper stochastic integrals with respect to Levy processes

    • Author(s)
      K.Sato
    • Journal Title

      Latin American Journal of Probability and Matheamtical Statistics (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Infinite divisibility for stochastic processes and time change

    • Author(s)
      O.E.Barndorff-Nielsen, M.Maejima, K.Sato
    • Journal Title

      J. Theoretical Probability (to appear)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Moments and projections of semistable probability measures on p-adic vector spaces

    • Author(s)
      M.Maejima, R.Shah
    • Journal Title

      J. Theoretical Probability (to appear)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Two families of improper stochastic integrals with respect to Levy processes

    • Author(s)
      K.Sato
    • Journal Title

      Latin American Journal of Probability and Matheamtical Statistics (to appear)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Moments and projections of semistable probability measures on p-adic vector spaces

    • Author(s)
      M.Maejima, R.Shah
    • Journal Title

      J.Theoret. Probab. (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the mixing coefficients of piecewise monotonic maps

    • Author(s)
      J.Aaronson, H.Nakada
    • Journal Title

      Israel J.Math. (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the central limit theorem for non-archimedean Diophantine approximations

    • Author(s)
      E.Deligero, H.Nakada
    • Journal Title

      Manuscripta Math. (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Infinitely divisibility and generalized subexponentiality

    • Author(s)
      T.Shimura, T.Watanabe
    • Journal Title

      Bernoulli (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Infinite divisibility for stochastic processes and time change

    • Author(s)
      O.E.Barndorff-Nielsen, M.Maejima, K.Sato
    • Journal Title

      J.Theoret. Probab. (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Publications] M.Maejima, K.Sato: "Semi-Levy processes, semi-selfsimilar additive processes, and semi-stationary Ornstein-Uhlenbeck type processes"J.Math.Kyoto Univ.. (発表予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Maejima, K.Yamamoto: "Long-memory stable Ornstein-Uhlenbeck processes"Electric J.Probab.. Vol.8 no.19. 1-18 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Inoue, H.Nakada: "The modified Jacobi-Perron algorithm over Fq(X)^d"Tokyo J.Math.. Vol.26 no.2. 447-470 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Takahashi, Y.Tamura: "Homogenization problems on disconnected fractal sets on R"Tokyo J.Math.. (発表予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] J.Pedersen, K.Sato: "Cone-parameter convolution semigroups and their subordination"Tokyo J.Math.. Vol.26 no.2. 503-525 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Sato: "Stochastic integrals in additive processes and application to semi-Levy processes"Osaka J.Math.. (発表予定).

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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