2018 Fiscal Year Final Research Report
Study on operator valued free probability, random matrices and their applications
| Project/Area Number |
15K04923
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | Aichi University of Education |
|---|
Principal Investigator |
|
|---|
| Research Collaborator |
Yoshida Hiroaki
Collins Benoit
Hasebe Takahiro
Suzuki Ryoichi
Ueda Yuki
|
|---|
| Project Period (FY) |
2015-04-01 – 2019-03-31
|
| Keywords | 自由確率論 / 無限分解可能分布 / ランダム行列 |
|---|
| Outline of Final Research Achievements |
In this research we proved the followings:
(1) We proved that the normal distribution is freely selfdecomposable. It means that free convolution semigroup of normal distributions are unimodal. It has some corollaries. In its proof, we found a necessary and sufficient condition for freely selfdecomposable distributions.
(2) We introduce non-commutative point of view to outlier problem. Based on moment methods, we proved that asymptotic cyclic monotone independence appear in our RMM. For some explicit models, we compute concrete limiting distributions.
|
| Free Research Field |
確率論
|
| Academic Significance and Societal Importance of the Research Achievements |
ランダム行列理論は統計学, 通信, 量子情報理論, 機械学習理論に応用がある. 自由確率論は計算が困難な巨大なランダム行列のスペクトル分布を見積もる手法であり, その深化は応用上極めて重要である. 本研究課題でも一例であげると, 統計学上重要なアウトライヤーの問題を扱っているこれはデータ解析に関連する話題であり, 我々の結果は具体的なモデルに対してアウトライヤーの位置を理論的に求める手法であり, それを統計学に応用すれば新しい検定手法などを構築でき, データ解析に応用できる可能性を秘めていると思われる.
|
