Synthetic Research of Probability Theory

我们的目的是通过我们各领域专家的研究成果的总和，为日本概率论社会做出贡献，以发展和利用概率论原有的遍历性。主办或协办国际会议4次（概率论与数论、大范围相互作用系统、数学金融2次），国内学术研讨会26次，资助参会人员285人，其中境外参会人员13人。我们得到了很多结果。其中，我们在此提出由首席研究员获得的3项结果。(1)通过将蒙特卡罗方法形式化为随机博弈（赌博），并应用柯尔莫哥罗夫的随机数理论，揭示了蒙特卡罗方法中抽样的本质问题。(2)利用多项式环的线性补全中的一个扩展遍历定理，计算了任意有限域上两个单多项式为互素的概率。(3)考虑一个随机的digamma函数，研究了它的中心极限定理，其结果可以看作是由半线上随机位置的电荷引起的涨落势。

Our purpose is to contribute to Japanese society of probability theory by the total sum of research results of our investigators, who are experts of each field, in order to develop and make use of the traversality that probability theory originally has.We sponsored or co-sponsored 4 international conferences (Probability and Number theory, Large Scale Interaction Systems, 2 conferences on Mathematical Finance), 26 domestic symposia, in which we financially supported 285 participants including 13 from overseas.We got many results. Among them, we here present 3 results obtained by the head investigator. (1) By formulating the Monte Carlo method as stochastic game (gambling), and applying Kolmogorov's random number theory, we revealed the essential problem of sampling in the Monte Carlo method (2) The probability of two monic polynomials over any finite fields to be coprime is computed by an extended ergodic theorem in the adelic completion of the ring of polynomials. (3) We considered a randomized digamma function, and investigated the central limit theorem for them, whose results can be regarded as the fluctuated potential caused by randomly located electric charges on the half line.