Assuming that {(Un,Vn)} is a sequence of càdlàg processes converging in distribution to (U,V) in the Skorohod topology, conditions are given under which {∬fn(β,u,v)dUndVn} converges weakly to ∬f(β,x,y)dUdV in the space C(R), where fn(β,u,v) is a sequence of “smooth” functions...

We study the joint limit distribution of the k largest eigenvalues of a p×p sample covariance matrix XXT based on a large p×n matrix X. The rows of X are given by independent copies of a linear process, Xit=∑jcjZi,t−j, with regularly varying noise (Zit) with tail index α∈(0,4). It is...

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.

We consider estimates motivated by extreme value theory for the correlation parameter of a first-order autoregressive process whose innovation distribution F is either positive or supported on a finite interval. In the positive support case, F is assumed to be regularly varying at zero, whereas...

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...

We show that the finite-dimensional distributions of a GARCH process are regularly varying, i.e., the tails of these distributions are Pareto-like and hence heavy-tailed. Regular variation of the joint distributions provides insight into the moment properties of the process as well as the...

Let Mt be the maximum of a recurrent one-dimensional diffusion up till time t. Under appropriate conditions, there exists a distribution function F such that P(Mt[less-than-or-equals, slant]x) - Ft(x)--0as t and x go to infinity. This reduces the asymptotic behavior of the maximum to that of the...

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...

We study the problem of estimating autoregressive parameters when the observations are from an AR process with innovations in the domain of attraction of a stable law. We show that non-degenerate limit laws exist for M-estimates if the loss function is sufficiently smooth; these results remain...