Stein's method is a remarkable and largely unexplored technique for proving approximation and limit theorems in probability, and the investigator hopes to develop versions of it which can be used to solve problems of interest to researchers in algebra, probability, and statistics. More precisely, there are five specific problems of interest the investigator would like to solve. The first problem is to use Stein's method to prove approximations to the Tracy-Widom distribution (whose merit lies in its applications to random matrix theory, tiling problems, percolation, queuing theory, and much else). The second problem is to continue the investigator's work on Stein's method and the spectrum of random walks. The third problem is to use Stein's method to prove approximations to the semicircle distribution (the free probability analog of the normal distribution). The fourth problem is to use Stein's method to bound convergence rates of Markov chains, building on ideas of Diaconis. The fifth problem is to prove chi-squared approximation for contingency tables with fixed row and column sums, which would be quite valuable because such approximations are frequently used in statistical work even though their accuracy is poorly understood.
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 plans to study the use of Stein's method in Markov chain theory, which as mentioned above, is crucial to drawing reliable conclusions from computer simulations.
