Probabilistic Properties and Statistical Inference for Regularly Varying Time Series

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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. ********