Spectral analysis of stochastic processes and random fields

随机过程及其应用是概率论的一个子领域，处理对真实生活数据的建模，表示随机值系统随时间的演变。常见的随机过程示例包括股市波动、音频和视频等信号、患者血压或体温等医疗数据以及随机移动(如随机行走)。广义的随机场描述了时间和空间的演变。随机场的例子包括静态图像、随机地形(风景)、波浪或非均质材质的组成变化。本研究致力于利用新技术研究随机过程或随机场变量之间的相互作用，从而更好地理解随机过程或随机场中的依赖程度。这些技术基于与其他数学领域中出现的概念的惊人关系，例如代数和动力系统。更具体地说，在许多情况下，变量之间的相互作用被视为时间序列相关性强度的度量。关于这些相互作用的信息是由一个被称为“谱密度函数”的函数给出的，该函数在时间序列和随机场理论中起着核心作用。数学中的其他谱概念有：“算子谱测量”，它生成马尔可夫过程的转变，以及从随机场中选择条目的大型随机矩阵的特征值的“经验谱分布”。这一研究项目旨在建立几个结构关系，在这三个明显不同的概念之间架起桥梁，每个概念的定义中都有“光谱”这个词。这项研究的结果将表明，这不是巧合。事实上，它们是相互关联的，它们相互揭示。这些关系将有助于确定随机进化中的依赖强度。他们将通过指出估计平稳随机过程和场的谱密度的新方法来扩展和改进用于分析平稳随机过程和场的工具。其中一些结果将有助于更好地理解具有相关项的随机矩阵的极限经验特征值分布，表现出长范围相关性。这项研究的目的是证明这种极限分布是由场的谱密度唯一确定的，并找到一个与它们相关的公式。

Stochastic processes and their applications are a subfield of probability theory dealing with modeling real life data, representing the evolution of a system of random values over time. Familiar examples of stochastic processes include stock market fluctuations, signals such as audio and video, medical data such as a patient's blood pressure or temperature, and random movement, such as random walk. A generalization, the random field, describes evolution in time and space. Examples of random fields include static images, random terrain (landscapes), waves, or composition variations of a heterogeneous material. This research project is concerned with a better understanding of the degree of dependence in a stochastic process or random field by using new techniques for studying the interactions between its variables. These techniques are based on surprising relationships with concepts that appear in other fields of mathematics, such as algebra and dynamical systems. The results will lead to a more accurate prediction of future values of random evolution.More specifically, in many situations, the interactions between the variables are viewed as a measure of the strength of dependence of a time series. The information about these interactions is given by a function called the "spectral density function," which plays a central role in the theory of time series and random fields. Other spectral notions in mathematics are: "operator spectral measure," which generates the transitions of a Markov process, and "empirical spectral distribution" of the eigenvalues of large random matrices with entries selected from random fields. This research project aims to establish several structural relationships that will build bridges between these three, apparently disparate notions, each having the word "spectral" in its definition. The results of this research will show that this is not a coincidence. As a matter of fact, they are interconnected and they shed light on one another. These relationships will facilitate determination of the strength of dependence in a random evolution. They will extend and improve tools used for analyzing stationary stochastic processes and fields by indicating new ways to estimate their spectral density. Some of the results will contribute to a better understanding of the limiting empirical eigenvalue distribution for random matrices with correlated entries, exhibiting long range dependence. The research aims to show that this limiting distribution is uniquely determined by the field's spectral density and to find a formula relating them.