单位环面上随机点的数方差的渐近行为
On the asymptotic behavior of the number variance of random points on the unit torus
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中文总结 AI 辅助
本文针对单位环面上均匀分布的独立同分布随机点，利用Komlós–Major–Tusnády定理与Karhunen-Loève分解，计算了数方差的分布并确定了泊松渐近失效的精确阈值。
中文摘要 AI 辅助
对单位环面上实序列分布的研究具有悠久且丰富的历史。存在多种描述给定确定性序列（通常具有算术起源）“伪随机”行为的概念，例如均匀分布（描述“全局尺度”上的分布）或对关联（一种“局部尺度”统计量）。另一种流行且重要的统计量是数方差$V_N(s)$，它可应用于从全局、中间到局部的所有尺度范围，描述当在环面上移动长度为$s$的区间时，序列初始段中该区间内元素数量的均方波动。在量子混沌学的术语中，这类统计量用于描述系统能级的分布，伪随机行为被称为泊松行为。由于数方差是衡量伪随机行为的重要指标，令人惊讶的是，文献中似乎并未包含关于真正随机情况（独立同分布随机点）下数方差的分布及几乎必然渐近行为的精确描述，即使在相对简单的单位环面上均匀分布点的设定中也是如此。在本文中，我们在参数$s$的宽范围内高精度地计算了数方差的分布，并确定了对于随机序列的典型实现，“泊松”渐近$V_N(s) \backsim N s$失效的精确阈值（以$s$表示）。关键技术输入来自Komlós–Major–Tusnády定理和Karhunen-Loève分解。
英文摘要
The study of the distribution of real sequences in the unit torus has a long and rich history. There are many notions which describe the ``pseudo-random'' behavior of a given deterministic sequence (often of arithmetic origin), such as equidistribution (which describes the distribution on a ``global scale'') or the pair correlation (which is a ``local-scale'' statistics). Another popular and important statistics, which can be applied to all ranges from global via intermediate to local, is the number variance $V_N(s)$, which describes the mean-square fluctuation of the number of elements of an initial segment of the sequence in an interval of length $s$, when moving this interval around the torus. In the terminology of quantum chaology, where such statistics are used to describe the distribution of the energy levels of a system, pseudo-random behavior is called Poissonian behavior. Since the number variance is an important indicator to qualify pseudo-random behavior, it is quite surprising that the literature does not seem to contain a precise description of the distribution and of the almost sure asymptotic behavior of the number variance in the truly random case (i.i.d.\ random points), even in the comparatively simple setup of uniformly distributed points on the unit torus. In the present paper we calculate the distribution of the number variance with high precision throughout a wide range of the parameter $s$, and determine the precise threshold (in terms of $s$) where the ``Poissonian'' asymptotics $V_N(s) \sim N s$ breaks for a typical realization of a random sequence. Key technical input comes from the Komlós--Major--Tusnády theorem and the Karhunen-Loève decomposition.
发表机构
- TU Graz（格拉茨工业大学）
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