We consider a simple bilinear process Xt=aXt-1+bXt-1Zt-1+Zt, where (Zt) is a sequence of iid N(0,1) random variables. It follows from a result by Kesten (1973, Acta Math. 131, 207-248) that Xt has a distribution with regularly varying tails of index [alpha]0 provided the equation Ea+bZ1u=1 has...

The goal of this paper is two-fold: (1) We review classical and recent measures of serial extremal dependence in a strictly stationary time series as well as their estimation. (2) We discuss recent concepts of heavy-tailed time series, including regular variation and max-stable processes.

Many real-life time series exhibit clusters of outlying observations that cannot be adequately modeled by a Gaussian distribution. Heavy-tailed distributions such as the Pareto distribution have proved useful in modeling a wide range of bursty phenomena that occur in areas as diverse as finance,...

A limit theory was developed in the papers of Davis and Dunsmuir (1996) and Davis et al. (1995) for the maximum likelihood estimator, based on a Gaussian likelihood, of the moving average parameter in an MA(1) model when is equal to or close to 1. Using the local parameterization, , where is the...

Davis and Mikosch (2009a) introduced the extremogram as a flexible quantitative tool for measuring various types of extremal dependence in a stationary time series. There we showed some standard statistical properties of the sample extremogram. A major difficulty was the construction of credible...

We use point processes theory to describe the asymptotic distribution of all upper order statistics for observations collected at renewal times. As a corollary, we obtain limiting theorems for corresponding extremal processes.

In the early 1990s, Avram and Taqqu showed that regularly varying moving average processes with all coefficients nonnegative and the tail index α strictly between 0 and 2 satisfy the functional limit theorem. They also conjectured that an equivalent statement holds under a certain less...

Extreme values of a stationary, multivariate time series may exhibit dependence across coordinates and over time. The aim of this paper is to offer a new and potentially useful tool called tail process to describe and model such extremes. The key property is the following fact: existence of the...

In this paper we give the theoretical basis of a possible explanation for two stylized facts observed in long log-return series: the long range dependence (LRD) in volatility and the integrated GARCH (IGARCH). Both these effects can be theoretically explained if one assumes that the data is...