Probabilistic Properties and Statistical Inference for Regularly Varying Time Series

概率和统计学中的一个经典假设是数据（随机变量）是独立的，近似高斯分布。然而，在许多实际应用中（如金融或保险），观察到的数据是依赖的（弱或强）和重尾的，即大的正值或负值发生的概率比高斯情况下更高。此外，极值观测往往形成集群，这意味着大的正值或负值出现在连续的时间点。因此，必须使用适合捕捉极端行为的时间序列模型。我的长期愿景是推进极端集群的概率理论，并利用它来开发相关的重尾数据的统计技术，目的是回答如何估计极端特征和集群指数的非常实际的问题。* * 1)簇的概率性质。* 我的目标是为重尾时间序列开发一个大观测集群的概率描述。该研究计划将涉及三个子课题：*1A）弱相关下的聚类性质;*1B）强相关下的聚类性质;*1C）具有极值独立性的时间序列。* 这些子主题中的每一个都需要显著不同的数学方法，然而，诸如规则变分、无限维空间上的测度的弱收敛和模糊收敛等高级工具将发挥重要作用。配备这些工具，我将获得新的令人兴奋的结果，将加深我们的时间序列的极端行为的理解。一些新的概念，如尾部过程和集群之间的联系将被开发。**2）聚类的统计推断。* 即使在独立的情况下，对极端情况的统计推断也提出了许多理论和实践挑战。我的目标是在弱依赖、强依赖和极端独立的背景下解决这些具有挑战性的问题，补充提案第一部分中的概率发展。我将使用来自经验过程理论和再培训技术的先进工具。因此，我将进一步发展统计理论和方法来估计极值特征和风险措施，扩展或简化许多现有的极限定理，并提供新的结果。此外，配备的概率工具，在研究计划的第一部分，我将建立新的方法来估计集群指数的情况下，弱相关。这是一个全新的研究方向。估计量的极限理论将用相应的恢复技术加以补充。因此，我的研究不仅会引起学术界的兴趣，也会引起从业者的兴趣，他们将能够使用我的软件进行统计分析。********

A classical assumption in probability and statistics is that data (random variables) are independent and approximately Gaussian. However, in many practical applications (like finance or insurance), the observed data are dependent (weakly or strongly) and heavy-tailed, that is, large positive or negative values occur with a higher probability than in a Gaussian case. Furthermore, extremal observations tend to form clusters, meaning that large positive or negative values occur at consecutive time points. As such, time series models suitable to capture extremal behaviour have to be used. My long-term vision is to advance probabilistic theory for clusters of extremes and use it to develop statistical techniques for dependent, heavy-tailed data, with the aim of answering very practical questions on how to estimate extremal characteristics and cluster indices. ******In this program, I will focus on the following aspects of time-series extremes.******1) Probabilistic properties of clusters. ***My goal is to develop a probabilistic description of clusters of large observations for heavy-tailed time series. The research program will involve three subtopics: ***1A) Cluster properties under weak dependence;***1B) Cluster properties under strong dependence;***1C) Time series with extremal independence. ***Each of these subtopics requires significantly different mathematical approaches, however, such advanced tools as regular variation, weak and vague convergence of measures on infinite dimensional spaces will play a major role. Equipped with these tools, I will obtain new exciting results that will deepen our understanding of extremal behaviour of time series. Several new concepts, like a link between a tail process and clusters will be developed. ******2) Statistical inference for clusters. ***Statistical inference for extremes presents many theoretical and practical challenges even in the case of independence. My goal is to tackle these challenging problems in the context of weak dependence, strong dependence as well as extremal independence, complementing the probabilistic development in the first part of the proposal. I will use advanced tools from the theory of empirical processes and resampling techniques. As such, I will further develop statistical theory and methodology for estimation of extremal characteristics and risk measures, extending or simplifying many existing limit theorems and providing new results. Furthermore, equipped with the probabilistic tools developed in the first part of the research program, I will establish new methodology for estimation of cluster indices in case of weak dependence. This is a completely new research direction. The limiting theory for estimators will be complemented with the corresponding resampling techniques. As such, my research will be not only of interest to academics, but also to practitioners, who will be able to use my software to perform statistical analysis. ********