Synthetic Research of Probability Theory

我们的目的是为了开发和利用概率论原本具有的横向性，将各领域的专家研究者的研究成果的总和贡献给日本概率论社会。我们主办或共同主办了4次国际会议（概率论与数论、大规模相互作用系统、2次数学金融会议），26次国内专题讨论会，我们资助了285名参与者，其中包括13名海外参与者。我们取得了很多成果。其中，我们在这里提出了3个结果，由首席研究员获得。(1)通过将MonteCarlo方法表示为随机博弈，并应用Kolmogorov随机数理论，揭示了MonteCarlo方法中抽样的本质问题。（2）利用多项式环的顶点完备化中的一个推广的遍历定理，计算了任意有限域上两个monic多项式互质的概率。(3)考虑了一类随机双伽玛函数，研究了它们的中心极限定理，其结果可以看作是半直线上随机分布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.