| Co-Investigator(Kenkyū-buntansha) |
TAKANOBU Satoshi Kanazawa Univ. Graduate Sch. Nat. Sci., Ass. Prof., 大学院・自然科学研究科, 助教授 (40197124)
OGAWA Shigeyoshi Kanazawa Univ. Dept. of Eng., Prof., 工学部, 教授 (80101137)
TANIGUCHI Setsuo Faculty of Mathematics, Kyushu University, Prof., 大学院・数理研究院, 教授 (70155208)
SATO Yoshihiro GIFU-Syotoku Gakuen Univ., Dept. of Economic Information, Ass. Prof., 経済情報学部, 助教授 (30249213)
FUKAI Yasunari Faculty of Mathematics, Kyushu University, Assistant, 大学院・数理研究院, 助手 (00311837)
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| Research Abstract |
1. A fundamental inequality for random sampling methods : A fundamental inequality for random sampling methods, which shows the limitation of accuracy of each method, has been obtained. According to the inequality, if we admit very complicated integrands, for any random sampling method, there exists an integrand for which the sampling method is as good as the i.i.d.-sampling or worse than it. In this sense, the i.i.d.-sampling is a stable numerical integration method for any complicated integrands.
2. Development of the random Weyl sampling : The random Weyl sampling, a random sampling method which requires much less randomness than i.i.d.-sampling, but is applicable for any complicated integrands, has been developed. It is a method by means of pairwise independent random samples. This reduction of randomness has the following advantages : (1) It is insensitive to the quality of pseudo-random generators. So, we can be careless in choosing pseudo-random generators. (2) The speed of generating samples is very rapid and it is almost independent of the speed of the pseudo-random generator in use. So, we can use a precise but slow pseudo-random generator, and then, a very reliable numerical integration for complicated integrands is possible.
3. Improvement of number theoretic density theorems by means of adeles : Number theoretic density theorems are not probabilistic objects by themselves. But we discovered that by using the ring of finite integral adeles(, which is an infinite dimensional extension of the ring of integers,) and the natural uniform probability measure on it, those density theorems can be formulated as proper probabilistic theorems, that is, law of large numbers. Moreover, we succeeded in describing the limit distributions of central limit theorem-scaled functions for a certain example.
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