2000 Fiscal Year Final Research Report Summary
A study on semi-selfdecomposable distributions and semi-selfsimilar stochastic processes
| Project/Area Number |
10440033
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| Research Category |
Grant-in-Aid for Scientific Research (B).
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Keio University |
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Principal Investigator |
MAEJIMA Makoto Keio University Mathematics Professor, 理工学部, 教授 (90051846)
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| Co-Investigator(Kenkyū-buntansha) |
KAWAZOE Takeshi Keio University Fac.of Policy Management Professor, 総合政策学部, 教授 (90152959)
TAMURA Yozo Keio University Mathematics Associate Professor, 理工学部, 助教授 (50171905)
NAKADA Hitoshi Keio University Mathematics Professor, 理工学部, 教授 (40118980)
WATANABE Toshiro The Univ. of Aizu Center for Mathematical Sciences Assistant Professor, 総合数理科学センター, 専任講師 (50254115)
SATO Ken-iti Nagoya Univ. Professor emeritus, 名誉教授 (60015500)
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| Project Period (FY) |
1998 – 2000
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| Keywords | infinitely divisible distributions / selfdecomposable distributions / semi-selfdecomposable distributions / type G distributions / selfsimilar stochastic processes / semi-selfsimilar stochastic processes / levy processes / absolute continuity |
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| Research Abstract |
1. The structure of the classes of semi-selfdecomposable distributions and its nested subclasses were clarified among the class of all infinitely divisible distributions. We introduced a way of making a new class of limitinz distributions derived from a class of distributions by taking the limit through some subsequence of normalized partial sums of independent random variables. We characterized completely a sort of a fixed point of this procedure.
2. In contrast to the absolute continuity of all selfdecomposable distributions, we found that semi-selfdecomposable distributions are not necessarily absolutely continuous. We also found a subclass of semi-selfdecomposable distributions which are always absolutely continuous.
3. We constructed some examples of non-selfdecomposable (or non-semi-selfdecomposable) distributions whose projections to lower dimensional spaces are selfdecomposable (or semi-selfdecomposable). This property has a sharp contrast to stable distributions.
4. We found that the marginal distributions of semi-selfsimilar processes at a time is selfdecomposable. We also found that their joint distributions at several times are closely related to nested subclasses of selfdecomposable distributions. It is also proved that similar observations remain true between selfdecomposable distributions and selfsimilar processes.
5. We succeeded in defining multivariate type G distributions and found a necessary and sufficient condition for that they are selfdecomposable. We also defined a sequence of nested subclasses of type G distributions and succeeded in making a new refinement of the class of infinitely divisible distributions.
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