Research on classical and non-commutative probability

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。该项目涉及随机场、随机矩阵、非对易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。为了调和经典中心极限定理和自由中心极限定理，他将研究一个非对易中心极限定理，该定理是在他以前的工作中通过例子提出的。他将研究统计的指数族和柯西-斯蒂尔杰斯核族之间的相似性，以此来将正态、二项、伽马和泊松定律与它们在自由概率上的对应关系联系起来。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带来新的矩阵模型。PI将使用Askey-Wilson多项式的正交性度量代替上述经典定律来构造具有线性回归和区间二次条件方差的马尔可夫过程。他还将分析描述几何量的大偏差行为的马尔可夫链，如指定次数的顶点数量在随机树的演化过程中的变化。这项研究源于对具有线性回归和二次条件方差的随机过程的研究。这些过程模拟了当初始和最终终点平均沿着一条直线给定时随机演变的现象。随机性作为与具有作为端点的二次函数的方差的线的偏差而出现。这样的过程被证明是马尔可夫过程，这并不令人惊讶；但令人惊讶的是，它们与非对易概率，特别是通常作为大型随机矩阵谱的近似而出现的自由概率表现出密切的联系。因此，PI还将研究非对易环境中的随机矩阵、二次回归问题和中心极限定理。另一个要研究的主题是随时间演变的随机树模型中出现的罕见现象。随机树在生物学、心理学和计算机科学中用作模型。罕见的感兴趣事件由偏离平衡的异常大的偏差组成，它们模拟了系统中罕见但有影响的行为。

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.