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Development of stochastic numerics and probability theory via lacunary series

Research Project

Project/Area Number 17340029
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 InstitutionKobe University

Principal Investigator

FUKUYAMA Katusi  Kobe University, 大学院理学研究科, 教授 (60218956)

Co-Investigator(Kenkyū-buntansha) HIGUCHI Yasunari  神戸大学, 大学院理学研究科, 教授 (60112075)
YAMAZAKI Tadashi  神戸大学, 大学院理学研究科, 教授 (30011696)
杉田 洋  大阪大学, 大学院・理学研究科, 教授 (50192125)
小川 重義  立命館大学, 理工学部, 教授 (80101137)
濱名 裕治  熊本大学, 大学院・自然科学研究科, 教授 (00243923)
安富 健児  立命館大学, 理工学部, 講師 (20388127)
Co-Investigator(Renkei-kenkyūsha) SUGITA Hiroshi  大阪大学, 大学院理学研究科, 教授 (50192125)
OGAWA Shigeyoshi  立命館大学, 理工学部, 教授 (80101137)
HAMANA Yuji  熊本大学, 大学院自然科学研究科, 教授 (00243923)
YASUTOMI Kenji  立命館大学, 理工学部, 専任講師 (20388127)
Project Period (FY) 2005 – 2008
Project Status Completed (Fiscal Year 2008)
Budget Amount *help
¥12,170,000 (Direct Cost: ¥10,700,000、Indirect Cost: ¥1,470,000)
Fiscal Year 2008: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2007: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2006: ¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2005: ¥3,200,000 (Direct Cost: ¥3,200,000)
Keywordsdiscrepancy / 間隙級数 / 重複対数の法則 / 中心極限定理 / Riesz-Raikov 和 / 再帰性 / Hardy-Littewood-Polya列 / 一様分布論 / 大間隙条件 / Riesz-Raikov和 / 概不変原理 / 間隙三角級数
Research Abstract

一様分布論にあらわれるdiscrepancy の漸近挙動の解析に取り組み、その重複対数の法則的な挙動が完全にわかっていなかった発散等比数列やいわゆるHardy-Littewood-P´olya の列について完全な形の重複対数が成立することを解明した。また、有界型重複対数の法則に従いながらも完全形の重複対数の法則が成り立たない例の構成などをおこない、一様分布論の測度的研究の進展に寄与した。

Report

(5 results)
  • 2008 Annual Research Report   Final Research Report ( PDF )
  • 2007 Annual Research Report
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (53 results)

All 2009 2008 2007 2006 2005 Other

All Journal Article (31 results) (of which Peer Reviewed: 30 results) Presentation (19 results) Book (3 results)

  • [Journal Article] A law of large numbers for arithmetic functions2009

    • Author(s)
      K. Fukuyama & Y. Komatsu
    • Journal Title

      Proceedings of the American Mathematical Society 137

      Pages: 349-352

    • NAID

      120005980788

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] On limit distribution of trigonometric sums2008

    • Author(s)
      K. Fukuyama & S. Takahashi
    • Journal Title

      Revue Roumaine des Mathematiques Pures et Appliques 53

      Pages: 19-24

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] On the central limit theorem and the law of the iterated logarithm2008

    • Author(s)
      K. Fukuyama & Y. Ueno
    • Journal Title

      Statistics & Probability Letters 78

      Pages: 1384-1387

    • NAID

      120005980790

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] The law of the iterated logarithm for subsequences: a simple proof2008

    • Author(s)
      K. Fukuyama & Y. Takeuchi
    • Journal Title

      Lobachevskii Jour. Math. 29

      Pages: 130-132

    • NAID

      120005980797

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] The law of the iterated logarithm for discrepancies of (θ^nx)2008

    • Author(s)
      K. Fukuyama
    • Journal Title

      Acta Mathematica Hungarica 118

      Pages: 155-170

    • NAID

      120005980770

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] The law of the iterated logarithm for discrepancies of {θ^nχ}2008

    • Author(s)
      K. Fukuyama
    • Journal Title

      Acta Mathematica Hungarica 118

      Pages: 155-170

    • Related Report
      2008 Annual Research Report 2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On limit distribution of trigonometric sums2008

    • Author(s)
      K. Fukuyama&S. Takahashi
    • Journal Title

      Revue Roumaine des Mathematiques Pures et Ap-pliques 53

      Pages: 19-24

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the central limit theorem and the law of the iterated logarithm2008

    • Author(s)
      K. Fukuyama&Y. Ueno
    • Journal Title

      Statistics&Probability Letters 78

      Pages: 1384-1387

    • NAID

      120005980790

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On recurrence property of Riesz-Raikov sums2007

    • Author(s)
      K. Fukuyama & R. Kondo
    • Journal Title

      Lobachevskii Jour. Math. 26

      Pages: 27-31

    • NAID

      120005980777

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] On lacunary trigonometric product2007

    • Author(s)
      K. Fukuyama
    • Journal Title

      Probability and Number Theory - Kanazawa 2005 49

      Pages: 79-90

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Analyticity of survival probability for contact process around t=02007

    • Author(s)
      V. Belitsky, Y. Higuchi, N. Konno, N. Sugimine
    • Journal Title

      Analysis and its Applications. 5

      Pages: 67-76

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] The probability of two Fq-polynomials to be coprime2007

    • Author(s)
      H. Sugita & S. Takanobu
    • Journal Title

      Probability and number theory-Kanazawa 2005 49

      Pages: 455-478

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] On a real-time scheme for the estimation of volatility.2007

    • Author(s)
      S. Ogawa, K. Wakayama
    • Journal Title

      Monte Carlo Methods Appl. no. 2

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] M. Pontier Pricing rules under asymmetric information2007

    • Author(s)
      S. Ogawa
    • Journal Title

      ESAIM Probab. Stat. 11

      Pages: 80-88

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] On recurrence property of Riesz-Raikov sums2007

    • Author(s)
      K. Fukuyama & R. Kondo
    • Journal Title

      Lobachevskii Journal of Mathematics 26

      Pages: 27-31

    • NAID

      120005980777

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On lacunary trigonometric product2007

    • Author(s)
      K. Fukuyama
    • Journal Title

      Advanced Studies in Pure Mathematics 49

      Pages: 79-90

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Gap series and functions of bounded variation2006

    • Author(s)
      K. Fukuyama
    • Journal Title

      Acta Mathematica Hungarica 110

      Pages: 175-191

    • NAID

      120005980769

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Numerical integration for complicated functions and random sampling2006

    • Author(s)
      H. Sugita
    • Journal Title

      19

      Pages: 153-169

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] On the range of pinned random walks2006

    • Author(s)
      Y. Hamana
    • Pages
      329-357
    • NAID

      110004762687

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] A remark on the range of three dimensional pinned random walks2006

    • Author(s)
      Y. Hamana
    • Journal Title

      Kumamoto J. Math. 19

      Pages: 83-97

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] On a real-time scheme for the estimation of volatility2006

    • Author(s)
      S. Ogawa, K. Wakayama
    • Journal Title

      Mem. Inst. Sci. Engrg. Ritsumeikan Univ. 65

      Pages: 29-39

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Gap series and functions of bounded variation2006

    • Author(s)
      K.Fukuyama
    • Journal Title

      Acta Mathematica Hungarica 110

      Pages: 175-191

    • NAID

      120005980769

    • Related Report
      2006 Annual Research Report 2005 Annual Research Report
  • [Journal Article] Backwardation in Asian option prices, International Journal of Innovative Computing2005

    • Author(s)
      J. Akahori, K. Yasutomi, T. Yokota
    • Journal Title

      Informations and Control 1

      Pages: 581-593

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] On asymptotic behavior of the prices of Asian options2005

    • Author(s)
      Y. Hishida K. Yasutomi
    • Journal Title

      Asia-Pacific Financial Markets 12

      Pages: 289-306

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Limit theorems for the Wiener sausage2005

    • Author(s)
      Y. Hamana
    • Journal Title

      Sugaku Expositions 18 no.1

      Pages: 53-73

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Introduction to the discrete approximation of stochastic differential equations2005

    • Author(s)
      S. Ogawa
    • Journal Title

      Sugaku Expositions 18 no. 1

      Pages: 101-122

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] S. Ogawa Numerical solutions of stochastic differential equations and their applications2005

    • Author(s)
      S. Kanagawa
    • Journal Title

      Sugaku Expositions 18 no. 1

      Pages: 75-99

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] The law of the iterated logarithm for the discrepancies of a permutation of(n_kx)

    • Author(s)
      K. Fukuyama
    • Journal Title

      Acta Mathematica Hungarica

      Pages: 0-0

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] A metric discrepancy result for the Hardy-Littlewood-Polya sequences

    • Author(s)
      K. Fukuyama & K. Nakata
    • Journal Title

      Monatshefte fur Mathematik

      Pages: 0-0

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] A law of the iterated logarithm for discrepancies: non-constant limsup Monatshefte f¨ur Mathematik

    • Author(s)
      K. Fukuyama
    • Journal Title

      Monatshefte fur Mathematik

      Pages: 0-0

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Noncausal stochastic calculus revisited-around the so-called Ogawa integral. Advancesin deterministic and stochastic analysis

    • Author(s)
      S. Ogawa
    • Journal Title

      World Sci. Publ.

      Pages: 297-320

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Presentation] Hardy-Littlewood-P´olya の列のdiscrepancy が従う重複対数の法則について統計数学分科会2009

    • Author(s)
      福山克司、中田啓介
    • Organizer
      日本数学会2009 年年会東京大学
    • Place of Presentation
      一般講演
    • Year and Date
      2009-03-26
    • Related Report
      2008 Final Research Report
  • [Presentation] Hardy-Littlewood-Polyaの列のdiscrepancyが従う重複対数の法則について2009

    • Author(s)
      福山克司, 中田啓介
    • Organizer
      統計数学分科会, 日本数学会2009年年会
    • Place of Presentation
      東京大学
    • Year and Date
      2009-03-26
    • Related Report
      2008 Annual Research Report
  • [Presentation] On the law of the iterated logarithm for discrepacies of fθnxg, Rencontres Math´ematiques en Rouen 20072009

    • Author(s)
      K. Fukuyama
    • Organizer
      Universit ´e de Rouen, France
    • Place of Presentation
      招待講演
    • Related Report
      2008 Final Research Report
  • [Presentation] 数列fnkxg の並べ変えのdiscrepancyの漸近挙動について統計数学分科会2008

    • Author(s)
      福山克司
    • Organizer
      日本数学会2008 年秋季総合分科会東京工業大学
    • Place of Presentation
      一般講演
    • Year and Date
      2008-09-24
    • Related Report
      2008 Final Research Report
  • [Presentation] 正の加法的函数を重みとした可積分同分布独立確率変数列の大数の法則について2008

    • Author(s)
      福山克司、小松豊
    • Organizer
      統計数学分科会, 日本数学会2008 年年会近畿大学
    • Place of Presentation
      一般講演
    • Year and Date
      2008-03-24
    • Related Report
      2008 Final Research Report
  • [Presentation] On the law of the iterated logarithm for fnkxg: non-constant limsup2008

    • Author(s)
      K. Fukuyama
    • Organizer
      Random matrices, special functions and related topics 京都大学数理解析研究所
    • Place of Presentation
      招待講演
    • Related Report
      2008 Final Research Report
  • [Presentation] 大きい間隙を持つ数列fnkxg のdiscrepancy の従う重複対数の法則2007

    • Author(s)
      福山克司
    • Organizer
      統計数学分科会, 日本数学会2007 年秋季総合分科会東北大学
    • Place of Presentation
      一般講演
    • Year and Date
      2007-09-21
    • Related Report
      2008 Final Research Report
  • [Presentation] On the law of the iterated logarithm for discrepancies of {θ^nχ}2007

    • Author(s)
      K. Fukuyama
    • Organizer
      Rencontres Mathematiques en Rouen 2007
    • Place of Presentation
      Universite de Rouen, France
    • Year and Date
      2007-06-14
    • Related Report
      2007 Annual Research Report
  • [Presentation] 数列fθnxg のDiscrepancy が従う重複対数の法則について2007

    • Author(s)
      福山克司
    • Organizer
      統計数学分科会, 日本数学会2007 年年会埼玉大学
    • Place of Presentation
      一般講演
    • Year and Date
      2007-03-27
    • Related Report
      2008 Final Research Report
  • [Presentation] Analyticity of survival probablity for contact process around t = 02007

    • Author(s)
      杉峰伸明、V. Belitsky, 樋口保成, 今野紀雄
    • Organizer
      統計数学分科会,日本数学会2007 年年会埼玉大学
    • Place of Presentation
      一般講演
    • Year and Date
      2007-03-27
    • Related Report
      2008 Final Research Report
  • [Presentation] 数列fθnxg のDiscrepancy が従う重複対数の法則について2007

    • Author(s)
      福山克司
    • Organizer
      東京確率論セミナー、東京工業大学
    • Place of Presentation
      一般講演
    • Year and Date
      2007-02-05
    • Related Report
      2008 Final Research Report
  • [Presentation] On the law of the iterated logarithm for discrepacies of fθnxg2007

    • Author(s)
      K. Fukuyama
    • Organizer
      Workshop on number theory and probability
    • Place of Presentation
      京都大学数理解析研究所および国際高等研究所, 招待講演
    • Related Report
      2008 Final Research Report
  • [Presentation] 弱い連続率の下での間隙級数の重複対数の法則についてII2006

    • Author(s)
      福山克司
    • Organizer
      統計数学分科会, 日本数学会2006 年秋季総合分科会大阪市立大学
    • Place of Presentation
      一般講演
    • Year and Date
      2006-09-19
    • Related Report
      2008 Final Research Report
  • [Presentation] ランダムなdigamma 関数に関する極限定理統計数学分科会2006

    • Author(s)
      杉田洋、高信敏
    • Organizer
      日本数学会2006 年秋季総合分科会大阪市立大学
    • Place of Presentation
      一般講演
    • Year and Date
      2006-09-19
    • Related Report
      2008 Final Research Report
  • [Presentation] 間隙級数の再帰性について2006

    • Author(s)
      福山克司、近藤良太
    • Organizer
      日本数学会2006 年年会中央大学
    • Place of Presentation
      一般講演
    • Year and Date
      2006-03-27
    • Related Report
      2008 Final Research Report
  • [Presentation] On asymptotic propertiesof gap series, 8 June 20062006

    • Author(s)
      K. Fukuyama
    • Organizer
      Workshop On Probability With Applications, National Taiwan University, Taipei, Taiwan
    • Place of Presentation
      招待講演
    • Related Report
      2008 Final Research Report
  • [Presentation] 弱い連続率の下での間隙級数の重複対数の法則について2005

    • Author(s)
      福山克司
    • Organizer
      統計数学分科会,日本数学会2005 年秋季総合分科会岡山大学
    • Place of Presentation
      一般講演
    • Year and Date
      2005-09-19
    • Related Report
      2008 Final Research Report
  • [Presentation] On the law of the iterated logarithm for gap series , Department of Mathematics, Hokkaido Univ.2005

    • Author(s)
      K. Fukuyama
    • Organizer
      Harmonic Analysis and its Applications at Sapporo,Organizers: A.Miyachi (Tokyo Woman'sChristian Univ.), K.Tachizawa(HokkaidoUniv.)
    • Place of Presentation
      招待講演
    • Year and Date
      2005-08-22
    • Related Report
      2008 Final Research Report
  • [Presentation] On the law of the iterated logarithm for gap series.2005

    • Author(s)
      K. Fukuyama
    • Organizer
      International Conference on Probability and Number Theory2005 (P&NT05,June 20 - June 24 ,2005), Kanazawa, Japan
    • Place of Presentation
      招待講演
    • Year and Date
      2005-06-23
    • Related Report
      2008 Final Research Report
  • [Book] Proceedings of the International Conference held in Kanazawa, June 20-24, 20052007

    • Author(s)
      Shigeki Akiyama, Kohji Matsumoto,Leo Murata and Hiroshi Sugita
    • Publisher
      49. Mathematical Society of Japan
    • Related Report
      2008 Final Research Report
  • [Book] 確率過程入門2006

    • Author(s)
      西尾真喜子・樋口保成
    • Total Pages
      239
    • Publisher
      培風館
    • Related Report
      2008 Final Research Report
  • [Book] 統計力学相転移の数理2006

    • Author(s)
      黒田耕嗣・樋口保成
    • Total Pages
      224
    • Publisher
      培風館
    • Related Report
      2008 Final Research Report

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Published: 2005-04-01   Modified: 2016-04-21  

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