CAREER: Asymptotics of random processes and their applications

该提案旨在推进随机过程的统计理论，这些随机过程表现出长期依赖性和非线性等特征，并在这个令人兴奋的新领域教育统计学家和其他科学家。与独立性假设下的成熟理论相比，建立具有这些特征的过程的极限理论具有更大的挑战性。作者提出了一种强大的基于鞅的方法，研究了谱估计、经验过程、非参数估计等相关渐近问题。具有长期相关性和非线性的过程发生在各个领域，包括计算机网络、通信、金融、地质、水文、计量经济学和大气科学等。提案中发展的研究成果的应用将有助于测试和证明科学家在这些领域的主张。特别是，PI开发了统计方法来确定温度和臭氧序列的趋势，并为气象学家关于气候变化模式的主张提供了统计推理。

The proposal aims to advance statistical theory for randomprocesses that exhibit features like long-range dependence andnonlinearities, and to educate both statisticians and othersscientists in this new exciting area. Compared to thewell-developed theory under the independence assumption, it isconsiderably more challenging to establish a limit theory forprocesses with such features. The Principal Investigator proposesa powerful martingale based method and studies spectralestimation, empirical processes, nonparametric estimation andother related asymptotic problems for such processes.Processes with long-range dependence and nonlinearities occur invarious fields, including computer networks, communication,finance, geology, hydrology, econometrics and atmospheric scienceamong others. Applications of the research results developed inthe proposal would help test and justify claims made by scientistsin such fields. In particular, the PI develops statisticalmethodology to identify trends in temperature and ozone sequencesand provides statistical reasoning for meteorologists' claims onclimate change patterns.