Mathematical Sciences: Selected Large Sample Theory Problemsin Probability and Statistics

这项研究涉及四个领域的工作概率和 统计，自助，大偏差，随机分形，和 以集合为索引的经验过程。 关于Bootstrap 该领域将涉及大样本性质的调查， 一个一般的加权自助，其中包括非参数 和贝叶斯引导作为特例。 研究在大 偏差将围绕着一个概念， 部分和的泛函大偏差 随机分形 将在各种随机的经验增量中进行研究 流程. 尾经验过程的行为， 将着眼于应用程序进行研究， 突出的是建立一致性和渐近性 分布的各种多元密度和模式估计。 概率论为以下研究提供了基础： 统计 这项研究将包括开发 概率的新结果，以及它们在 统计理论

This research involves work in four areas of probability and statistics, the bootstrap, large deviations, random fractals, and empirical processes indexed by sets. Research in the bootstrap area will involve investigation of the large sample properties of a general weighted bootstrap, which includes the nonparametric and Bayesian bootstraps as special cases. The research in large deviations will be centered around a notion of extended functional large deviations for partial sums. Random fractals will be studied in empirical increments of various random processes. The behavior of tail empirical processes indexed by sets will be studied with an eye towards applications, the most prominent being in establishing the consistency and asymptotic distribution of various multivariate density and mode estimators. Probability theory provides a basis for research in statistics. This research will include both the development of new results in probability, as well as their application to statistical theory.