Mathematical Sciences: Quantile-Uniform Empirical Distribution Function Methods in Probability Theory

首席研究员将继续利用独立同分布随机变量序列的表示，作为在独立同分布均匀变量中观察到的分位数函数序列。分位数函数是分布函数的左连续倒数。这种表示在随机变量的部分和和经验过程之间提供了强有力的联系。在过去的几年里，首席研究员和许多合作者使用这种技术获得了许多有趣的结果。研究的具体问题包括加权逼近的Kiefer过程版本、Bahadur-Kiefer型过程的收敛性、阶统计量的加权和的收敛性以及极值和的迭代对数定律。

The principal investigator will continue to exploit the representation of a sequence of independent identically distributed random variables as the sequence of the quantile function observed at independent identically distributed uniform variables. The quantile function is the left continuous inverse of the distribution function. This representation provides a strong connection between the partial sums of random variables and the empirical process. The principal investigator, along with a number of collaborators, has used this technique over last several years to get many interesting results. Specific problems of study include the Kiefer process version of the weighted approximations, convergence of Bahadur-Kiefer type process, convergence of weighted sums of order statistics, and iterated logarithm laws for the sums of extreme values.