Stein's Method and its Applications

University Of Southern California, Los Angeles CA

Investigators

Abstract

Stein's method is a powerful technique for proving approximation and limit theorems in probability. The investigator hopes to develop versions of it which can be used to solve problems of interest to researchers in mathematical physics, spectral theory, and Markov chain theory. After completing works in progress related to Stein's method, the investigator will focus on four specific goals. The first goal is to use Stein's method to study determinantal point processes, which appear in random matrix theory, tiling problems, percolation, queuing theory, and elsewhere. The second goal is to continue the investigator's work on Stein's method and the spectrum of random walks, focusing on non-normal distributions, concentration inequalities, and convergence rates. The third goal is to use Stein's method to study error terms in Markov chain theory: central limit theorems and large deviations for Markov chains, hitting times, and the cutoff phenomenon. A fourth goal is to extend the scope of Stein's method to non-reversible settings. In making decisions of public policy, it is important that one draws correct conclusions from the data available. A drawback to many of the statistical methods applied today is that the accuracy of the conclusions is unclear. As one example, in determining whether there is gender bias in the promotion of employees, it is common to use a statistical test called the chi-squared test. As a second example, work on DNA sequences of interest to biotechnology and health uses Poisson approximations from probability theory. As a third example, in physics and chemistry one uses Markov chain methods to test theoretical models. These methods are routinely applied so it is important to quantify their validity. A remarkable technique, introduced by Charles Stein at Stanford, offers hope of better quantifying the accuracy of such procedures. Stein's method has proven successful in applications to DNA sequencing, but the technique is largely unexplored. The investigator plans to extend the scope and applicability of this method. As a final goal, the investigator will study the use of Stein's method in Markov chain theory, which as mentioned above, is crucial to drawing reliable conclusions from computer simulations.