Research on classical and non-commutative probability

经典概率和非交换概率研究

• 批准号: 0904720
• 负责人: Wlodzimierz Bryc
• 金额: $ 11.82万
• 依托单位: University of Cincinnati Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0904720&HistoricalAwards=false
• 关键词: Research; classical; non; commutative; probability

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The project involves research on conditional moments of random fields, random matrices, noncommutative probability, and large deviations.The PI will explore connections between classical and free probability from several inter-related angles. To reconcile the classical and free central limit theorems, he will study a noncommutative central limit theorem that was suggested by examples in his previous work. He will study similarities between the exponential families of statistics and the Cauchy-Stieltjes kernel families as a way to relate the normal, binomial, gamma and Poisson laws to their counterparts in free probability. He will investigate matrix ensembles that share similarities with such laws, and may lead to new matrix models for the corresponding free probability laws. The PI will use the orthogonality measures of the Askey-Wilson polynomials as a replacement for the above mentioned classical laws to construct Markov processes with linear regressions and quadratic conditional variances on an interval.He will also analyze large deviations of Markov chains that describe the behavior of geometric quantities like the number of vertexes of prescribed degree as they vary during the evolution of a random tree.This research originated from the study of random processes that have linear regressions and quadratic conditional variances. These processes model phenomena that evolve at random when the initial and final endpoints are given by following a straight line on average. The randomness occurs as deviations from that line with variance that is a quadratic function of the endpoints. Such processes, not surprisingly, turn out to be Markov; but surprisingly they exhibit intimate connections to noncommutative probability, and in particular to free probability that usually arises as approximation to spectra of large random matrices. Thus the PI will also investigate random matrices, quadratic regression problems, and the centrial limit theorem in a noncommutative setting. A separate topic to be investigated are rare phenomena arising in random tree models that evolve in time. Random trees serve as models in biology, psychology, and computer science.Rare events of interest consist of unusually large deviations from the equilibrium, and they model a rare but influential behavior of the system.

该奖项根据 2009 年美国复苏和再投资法案（公法 111-5）提供资金。该项目涉及随机场、随机矩阵、非交换概率和大偏差的条件矩的研究。PI将从几个相互关联的角度探索经典概率和自由概率之间的联系。为了协调经典和自由中心极限定理，他将研究他之前工作中的例子所提出的非交换中心极限定理。他将研究统计指数族与 Cauchy-Stieltjes 核族之间的相似性，以此将正态、二项式、伽玛和泊松定律与自由概率中的对应定律联系起来。他将研究与这些定律有相似之处的矩阵系综，并可能为相应的自由概率定律得出新的矩阵模型。 PI 将使用 Askey-Wilson 多项式的正交性度量来替代上述经典定律，以在区间上构建具有线性回归和二次条件方差的马尔可夫过程。他还将分析马尔可夫链的大偏差，这些偏差描述了几何量（如规定度数的顶点数量）在随机树演化过程中变化时的行为。这项研究 起源于对具有线性回归和二次条件方差的随机过程的研究。当初始和最终端点通过平均直线给出时，这些过程模拟随机演变的现象。随机性表现为与该线的偏差，其方差是端点的二次函数。 毫不奇怪，这样的过程被证明是马尔可夫的。但令人惊讶的是，它们表现出与非交换概率的密切联系，特别是与通常作为大型随机矩阵谱的近似而出现的自由概率。因此，PI 还将研究非交换环境中的随机矩阵、二次回归问题和中心极限定理。 要研究的一个单独主题是随时间演变的随机树模型中出现的罕见现象。 随机树充当生物学、心理学和计算机科学中的模型。感兴趣的罕见事件由与平衡的异常大的偏差组成，它们模拟了系统的罕见但有影响力的行为。