Infinite dimensional and numerical stochastic analysis

1.随机抽样方法的一个基本不等式：得到了随机抽样方法的一个基本不等式，它表明了每种方法的精度限制。根据不等式，如果我们允许非常复杂的被积函数，对于任何随机抽样方法，都存在一个被积函数，其抽样方法与独立同分布一样好。采样或更糟。从这个意义上说，i.i.d.-对于任何复杂的被积函数，抽样方法都是一种稳定的数值积分方法.随机外尔抽样的发展：随机外尔抽样，一种比i.i.d.要求更少随机性的随机抽样方法。采样，但适用于任何复杂的被积体，已经开发。它是一种利用两两独立随机样本的方法。这种随机性的降低具有以下优点：（1）它对伪随机发生器的质量不敏感。因此，我们可以在选择伪随机生成器时不小心。(2)生成样本的速度非常快，并且几乎与所使用的伪随机发生器的速度无关。因此，我们可以使用一个精确但缓慢的伪随机发生器，然后，一个非常可靠的数值积分复杂的被积体是可能的。用Adele改进数论密度定理：数论密度定理本身不是概率对象。但我们发现，利用有限积分环（是整数环的无限维推广）和其上的自然一致概率测度，这些密度定理可以表述为真概率定理，即大数定律。此外，我们还成功地描述了中心极限定理尺度函数的极限分布。

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.