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2006 Fiscal Year Final Research Report Summary

Study on limit theorems for fuzzy set-valued random variables

Research Project

Project/Area Number 17540123
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

OGURA Yukio  Saga University, Department of Mathematics, Professor, 理工学部, 教授 (00037847)

Co-Investigator(Kenkyū-buntansha) MATSUMOTO Hiroyuki  Nagoya University, Graduate School of Information Science, Professor, 大学院・情報科学研究科, 教授 (00190538)
SHIOYA Takashi  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90235507)
TOMISAKI Matsuyo  Nara Women's University, Department of Mathematics, Professor, 理学部, 教授 (50093977)
MITOMA Itaru  Saga University, Department of Mathematics, Professor, 理工学部, 教授 (40112289)
HANDA Kenji  Saga University, Department of Mathematics, Associate Professor, 理工学部, 助教授 (10238214)
Project Period (FY) 2005 – 2006
KeywordsMarkov property / generalized diffusion processes / fuzzy set-valued random variables / large deviation principles / Fell-Matheron topology / Choquet theorem / Choquet capacity / law of large numbers
Research Abstract

1. We gave a necessary and sufficient condition for stochastic processes in a class of one-dimensional Markovian (not necessarily strong Markovian) processes with continuous paths to be bi-generalized diffusion process which were introduced by the head investigator. It is worth to note that the scale functions of the processes in our class are no more continuous in general and may miss the strong Markov property at the discontinuous points of scales.
2. We gave large deviation principles for sums of independent identically distributed fuzzy set-valued random variables with compact level sets. More precisely, we gave Cramer type large deviation principles for such sums with respect to the topologies induced by the mean convergence of order p, Levy's metric and graph convergence, and the relative topology of a countable direct product topology.
3. We extended a Choquet theorem on the construction of set-valued random variables to that on the construction of fuzzy set-valued random variables. That is, we showed existence of a fuzzy set-valued random variable associated with an alternating Choquet capacity of infinite order with respect to the graph topology which coincides the relative topology of product Fell-Matheron topology.
4. We gave a concrete metric compatible with the Fell-Matheron topology on the space of closed sets in a locally compact second countable Hausdorff space. We had to modify Hausdorff-Buseman which is given in the book "Random sets" by I. Molchanov introduced as a compatible metric.
5. We gave a strong law of large numbers for sums of independent fuzzy set-valued random variables, which are not necessarily indentically distributed.

  • Research Products

    (32 results)

All 2007 2006 2005 Other

All Journal Article (31 results) Book (1 results)

  • [Journal Article] Asymptotic Expansion of Perturbative Chern-Simons Theory via Wiener Space2007

    • Author(s)
      Sergio Albeverio
    • Journal Title

      SFB, Bonn University 611・322

      Pages: 50

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Horizontal Lift of the Brownian motion on the hyperbolic plane and the Selberg trace formula2007

    • Author(s)
      H.Matsumoto
    • Journal Title

      Jour. Funct. Analysis 244

      Pages: 565-578

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On Some Metrics Compatible with the Fell-Matheron Topology2007

    • Author(s)
      Yukio Ogura
    • Journal Title

      Internat. Jour. Approximate Reasoning (Bonn University) 611 No.322

      Pages: 5O

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Exponential functionals of Brownian motion, I, Probability laws at fixed time2007

    • Author(s)
      Hiroyuki Matsumoto, Marc Yor
    • Journal Title

      Probability Surveys Vol. 244

      Pages: 565-578

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On a Choquet theorem for random upper semicontinuous functions2006

    • Author(s)
      Yukio Ogura
    • Journal Title

      Soft Method for Integrated Uncertainty Modelling

      Pages: 145-151

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A note on random upper semicontinuous functions2006

    • Author(s)
      Hung T. Nguen
    • Journal Title

      A note on random upper semicontinuous functions

      Pages: 129-135

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Strong law of large numbers for independent fuzzy set-valued random variables2006

    • Author(s)
      Shoumei Li
    • Journal Title

      Fuzzy Sets and Systems Vol. 157

      Pages: 2569-2578

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] ある反転公式とその応用2006

    • Author(s)
      半田賢司
    • Journal Title

      統計数理研究所共同研究リポート 1842

      Pages: 105-115

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Asymptotic conditional distributions related to one-dimensional generalized diffusion processes2006

    • Author(s)
      M.Iizuka
    • Journal Title

      Tsukuba Journal of Mathematics 30

      Pages: 273-327

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On a Choquet theorem for random upper semicontinuous functions2006

    • Author(s)
      Yukio Ogura
    • Journal Title

      Soft Method for Integrated Uncertainty Modelling (Springer)

      Pages: 145-151

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A note on random upper semicontinuous functions2006

    • Author(s)
      Hung T.Nguen, Yukio Ogura, Santi Tasena, Hien Tran
    • Journal Title

      Soft Method for Integrated Uncertainty Modelling (Springer)

      Pages: 129-135

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Strong law of large numbers for independent fuzzy set-valued random variables2006

    • Author(s)
      Shoumei Li, Yukio Ogura
    • Journal Title

      Fuzzy Sets and Systems Vol. 157

      Pages: 2569-2578

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Asymptotic conditional distributions related to one- dimensional generalized diffusion processes2006

    • Author(s)
      Masaru Iizuka, Miyuki Maeno, Matsuyo Tomisaki
    • Journal Title

      Tsukuba Journal of Mathematics Vol. 30, No.2

      Pages: 273-327

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Sampling formulae for symmetric selection2005

    • Author(s)
      K.Handa
    • Journal Title

      Electron. Commun. Probab. Vol. 10

      Pages: 223-234

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Exponential functionals of Brownian motion, I, Probability laws at fixed time2005

    • Author(s)
      H.Matsumoto
    • Journal Title

      Probability Surveys Vol. 2

      Pages: 312-347

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Exponential functionals of Brownian motion, II, Some related diffusion processes2005

    • Author(s)
      H.Matsumoto
    • Journal Title

      Probability Surveys Vol. 2

      Pages: 348-384

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Volume collapsed three-manifolds with a lower Curvature bound2005

    • Author(s)
      T.Yamaguchi
    • Journal Title

      Math. Ann. 333・1

      Pages: 131-155

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On a class of one-dimensional Markov processes with continuous paths2005

    • Author(s)
      Nubuyuki Ikeda, Yukio Ogura
    • Journal Title

      "Stochastic Analysis and Partial Differential Equations" AMS Contemp. Math. Vol. 10 (to appear)

      Pages: 223-234

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Asymptotic Expansion of Perturbative Chern-Simons Theory via Wiener Space2005

    • Author(s)
      Sergio Albeverio, Itaru Mitoma
    • Journal Title

      SFB Vol. 2

      Pages: 312-347

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Sampling formulae for symmetric selection2005

    • Author(s)
      Kenji Handa
    • Journal Title

      Electron. Commun. Probab. Vol. 2

      Pages: 348-384

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Exponential functionals of Brownian motion, II, Some related diffusion processes2005

    • Author(s)
      Hiroyuki Matsumoto, Marc Yor
    • Journal Title

      Probability Surveys Vol. 333, no. 1

      Pages: 131-155

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On Some Metrics Compatible with the Fell-Matheron Topology

    • Author(s)
      Yukio Ogura
    • Journal Title

      Internat. Jour. Approximate Reasoning (To appear)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On a class of one-dimensional Markov processes with continuous paths

    • Author(s)
      Nubuyuki Ikeda
    • Journal Title

      Stochastic Analysis and Partial Differential Equations" AMS Comtemp. Math. (To appear)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Variational convergence over metric spaces

    • Author(s)
      Kuzuhiro Kuwae
    • Journal Title

      Trans. of AMS (To appear)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Fixed point sets of parabolic isometries of CAT(O)-spaces

    • Author(s)
      K.Fujiwara
    • Journal Title

      Comment. Math. Helv. (To appear)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Inverse local times of harmonic transformed Bessel processes

    • Author(s)
      M.Tomisaki
    • Journal Title

      Ann. Reports of Graduate school of Humanities and Sciences Nara Women's University (To appear)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Horizontal lift of the Brownian motion on the hyperbolic plane and the Selberg trace formula

    • Author(s)
      Hiroyuki Matsumoto
    • Journal Title

      J. Func. Anal.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Volume collapsed three-manifolds with a lower curvature bound

    • Author(s)
      Takao Yamaguchi, Takashi Shioya
    • Journal Title

      Math. Ann.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Variational convergence over metric spaces

    • Author(s)
      Kazuhiro Kuwae, Takashi Shioya
    • Journal Title

      Trans. of Amer. Math. Soc (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Fixed point sets of parabolic isometries of CAT(0)-spaces

    • Author(s)
      Koji Fujiwara, Takashi Shioya, Koichi Nagano
    • Journal Title

      Comment. Math. Helv. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Inverse local times of harmonic transformed Bessel processes

    • Author(s)
      Matsuyo Tomisaki
    • Journal Title

      Ann. Reports of Graduate School of Humanities and Sciences Nara Women's University (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] 確率論入門I2006

    • Author(s)
      池田信行
    • Total Pages
      323+x
    • Publisher
      培風館
    • Description
      「研究成果報告書概要(和文)」より

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Published: 2008-05-27  

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