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Density theorem in number theory and limit theorem in probability theory - LLN, CLT etc.

数论中的密度定理和概率论中的极限定理 - LLN、CLT 等。

基本信息

批准号：
13640108
负责人：
TAKANOBU Satoshi
金额：
$ 2.11万
依托单位：
Kanazawa University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

项目摘要

Our purpose of the project is as follows : We formulate Dirichlet's density theorem stating the probability of two integers to be co-prime as law of large numbers (LLN), and then we consider central limit theorem scaling (CLT-scaling) and find a limit theorem on it.Let us consider (Zhat, ＼lambda) as a fundamental probability space, where Zhat is a finite integral adele and ＼lambda the Haar probability measure on it. For each (x,y) ＼in Zhat ＼times Zhat, let X(x,y) = 1 or 0 according as (x,y) is co-prime or not. Then, as N＼to＼infty, S_N(x,y) = (1/N)^2 ＼sum_{m,n=1}^N X(x+m,y+n) converges to 6/＼pi^2 a.s., which is just LLN.Next we consider the limit behavior of CLT-scaling N(S_N(x,y)-6/＼pi^2). Then we can describe completely the set of all limit points of {N(S_N(x,y)-6/＼pi^2)} in the L^2-space by parametrizing them continuously in terms of elements of a quotient ring Zhat/＼sim. In particular, N(S_N(x,y)-6/＼pi^2) is not convergent as N＼to＼infty. In a word, CLT does not hold!If, however, we … More interpret the convergence in the sense of Cesaro, then(1/N) ＼sum_{n=l}^N n(S_n(x,y}-6/＼pi^2) ＼to U(x) + U(y) in l^2.HereU(x) = ＼sum_{u=1}^{＼infty} (＼mu(u)/u) ((x＼mod u)/u - (u-1)/2u) in L^2,where ＼mu(u) is the Mobius function. So our study turns to an investigation of this U. In this project, it is seen that the distribution of U is symmetric and has moments of all orders. We further expect that U will be not normal distributed, although normal distributions with mean zero have the property above. If this is proved, we want to call the convergence above non CLT.On the one hand, from the description of limit points of {N(S_N(x,y) -6/＼pi^2)} N_k(S_{N_K}(x,y)-6/＼pi^2) ＼to 0 in L^2 for whatever subsequence {N_k} such that N_k ＼not=0 and N_k ＼to 0 in Zhat/＼sim. Renormalizing this by its standard deviation in order to find a nontrivial limit, we expect that the renormalization will converge to a standard normal distribution.We can not succeed in proving these two conjectures within the term of project. We are instead giving a verification by computational experiment. Less

我们的项目目的如下：我们将Dirichlet密度定理表述为大数定律（LLN），然后我们考虑中心极限定理标度（CLT-缩放）并找到其极限定理。让我们考虑（Zhat，\lambda）作为基本概率空间，其中Zhat是有限积分adele，\lambda是其上的Haar概率测度。对于每个（x，y）\in Zhat \times Zhat，设X（x，y）= 1或0，取决于（x，y）是否互质.然后，当N\to\infty时，S_N（x，y）=（1/N）^2 \sum_{m，n=1}^NX（x+m，y+n）收敛于6/\pi ^2 a.s.，接下来，我们考虑CLT缩放N（S_N（x，y）-6/\pi^2）的极限行为。然后我们可以完全描述{N（S_N（x，y）-6/\pi^2）}在L^2-空间中的所有极限点的集合，通过连续地用商环Zhat/\sim的元素参数化它们.特别地，N（S_N（x，y）-6/\pi^2）不收敛为N\to\infty。一句话，CLT不成立！然而，如果我们 ...更多信息在塞萨罗意义下解释收敛性，则在L^2上有（1/N）\sum_{n= 1}^N n（S_n（x，y）-6/\pi^2）\to U（x）+ U（y）.这里U（x）= \sum_{u=1}^{\infty}（\mu（u）/u）（（x\mod u）/u -（u-1）/2u），其中\mu（u）是Mobius函数.因此，我们的研究转向了对这一问题的调查。在这个项目中，可以看到U的分布是对称的，并且具有所有阶的矩。我们进一步预期U将不是正态分布，尽管均值为零的正态分布具有上述性质。如果证明了这一点，我们就把上述收敛称为非CLT收敛.一方面，由{N（S_N（x，y）-6/\pi^2）} N_k（S_{N_K}（x，y）-6/\pi^2）\to 0在L^2中的极限点的描述，对于任意的子列{N_k}，使N_k \not=0，N_k \to 0在Zhat/\sim中.通过标准差重正化以找到一个非平凡的极限，我们期望重正化将收敛到一个标准正态分布，但在项目期限内我们无法成功地证明这两个定理。相反，我们通过计算实验进行验证。少