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For stationary time series with regularly varying marginal distributions, an important problem is to estimate the associated tail index which characterizes the power‐law behavior of the tail distribution. For this, various results have been developed for independent data and certain types of dependent data. In this article, we consider the problem of tail index estimation under a recently proposed notion of serial tail dependence called the tail adversarial stability. Using the technique of adversarial innovation coupling and a martingale approximation scheme, we establish the consistency and central limit theorem of the tail index estimator for a general class of tail dependent time series. Based on the asymptotic normal distribution from the obtained central limit theorem, we further consider an application to cluster a large number of regularly varying time series based on their tail indices by using a robust mixture algorithm. The results are illustrated using numerical examples including Monte Carlo simulations and a real data analysis. 

Summary In this article we develop an asymptotic theory for sample tail autocorrelations of time series data that can exhibit serial dependence in both tail and non-tail regions. Unlike with the traditional autocorrelation function, the study of tail autocorrelations requires a double asymptotic scheme to capture the tail phenomena, and our results do not impose any restrictions on the dependence structure in non-tail regions and allow processes that are not necessarily strongly mixing. The newly developed asymptotic theory reveals a previously undiscovered phase transition phenomenon, where the asymptotic behaviour of sample tail autocorrelations, including their convergence rate, can transition from one phase to another as the lag index moves past the point beyond which serial tail dependence vanishes. The phase transition discovery fills a gap in existing research on tail autocorrelations and can be used to construct the lines of significance, in analogy to the traditional autocorrelation plot, when visualizing sample tail autocorrelations to assess the existence of serial tail dependence or to identify the maximal lag of tail dependence. 

Summary Quantile regression is a popular and powerful method for studying the effect of regressors on quantiles of a response distribution. However, existing results on quantile regression were mainly developed for cases in which the quantile level is fixed, and the data are often assumed to be independent. Motivated by recent applications, we consider the situation where (i) the quantile level is not fixed and can grow with the sample size to capture the tail phenomena, and (ii) the data are no longer independent, but collected as a time series that can exhibit serial dependence in both tail and non-tail regions. To study the asymptotic theory for high-quantile regression estimators in the time series setting, we introduce a tail adversarial stability condition, which had not previously been described, and show that it leads to an interpretable and convenient framework for obtaining limit theorems for time series that exhibit serial dependence in the tail region, but are not necessarily strongly mixing. Numerical experiments are conducted to illustrate the effect of tail dependence on high-quantile regression estimators, for which simply ignoring the tail dependence may yield misleading $$p$$-values.