Extremal Indices in the Series Scheme and their Applications

Joint Extremal Behavior of Hidden and Observable Time Series with an Application to GARCH Processes

We study the behavior of a real-valued and unobservable process (Y_t) under an extreme event of a related process (X_t) that is observable. Our analysis is motivated by the well-known GARCH model which represents two such sequences, i.e. the observable log returns of an asset as well as the hidden volatility process. Our results complement the findings of Segers (2007) and Smith (1992) for a single time series. We show that under suitable assumptions their concept of a tail chain as a limiting process is also applicable to our setting. Furthermore, we discuss existence and uniqueness of a limi…

Extremes of scale mixtures of multivariate time series

Factor models have large potencial in the modeling of several natural and human phenomena. In this paper we consider a multivariate time series $\mb{Y}_n$, ${n\geq 1}$, rescaled through random factors $\mb{T}_n$, ${n\geq 1}$, extending some scale mixture models in the literature. We analyze its extremal behavior by deriving the maximum domain of attraction and the multivariate extremal index, which leads to new ways to construct multivariate extreme value distributions. The computation of the multivariate extremal index and the characterization of the tail dependence show the interesting prope…

Large deviations of lp-blocks of regularly varying time series and applications to cluster inference

In the regularly varying time series setting, a cluster of exceedances is a short period for which the supremum norm exceeds a high threshold. We propose to study a generalization of this notion considering short periods, or blocks, with lp-norm above a high threshold. Our main result derives new large deviation principles of extremal lp-blocks, which guide us to define and characterize spectral cluster processes in lp. We then study cluster inference in lp to motivate our results. We design consistent disjoint block estimators to infer features of cluster processes. Our estimators promote the…

A multivariate model for financial indices and an algorithm for detection of jumps in the volatility

We consider a mean-reverting stochastic volatility model which satisfies some relevant stylized facts of financial markets. We introduce an algorithm for the detection of peaks in the volatility profile, that we apply to the time series of Dow Jones Industrial Average and Financial Times Stock Exchange 100 in the period 1984-2013. Based on empirical results, we propose a bivariate version of the model, for which we find an explicit expression for the decay over time of cross-asset correlations between absolute returns. We compare our theoretical predictions with empirical estimates on the same…

鞅变换的一些极值问题，I

Some extremal problems for martingale transforms, I

本文研究有界鞅的鞅变换分布估计的极值问题，通过构造满足边界条件的逐点最小对角凹Bellman函数，在非对称边界和窄带条件下给出了解。

Multivariate maxima of moving multivariate maxima

We define a class of multivariate maxima of moving multivariate maxima, generalising the M4 processes. For these stationary multivariate time series we characterise the joint distribution of extremes and compute the multivariate extremal index. We derive the bivariate upper tail dependence coefficients and the extremal coefficient of the new limiting multivariate extreme value distributions.