A simple procedure is proposed for estimating the coefficients {[psi]} from observations of the linear process X1=[summation operator]xJ=0[psi]JZ1-j, 1=1,2... The method is based on the representation of X1 in terms of the innovations, Xn-Xn, N=1,..., 1, where Xn is the best mean square...

Let Xt = [Sigma][infinity]j=-[infinity] cjZt - j be a moving average process where {Zt} is iid with common distribution in the domain of attraction of a stable law with index [alpha], 0 [alpha] 2. If 0 [alpha] 2, EZ1[alpha] [infinity] and the distribution of Z1and Z1Z2 are tail equivalent...

Let {Zn} be an iid sequence of random variables with common distribution F which belongs to the domain of attraction of exp{-e-x}. If in addition, F[epsilon]Sr([gamma]) (i.e.,limx--[infinity] P[Z1+Z2]/P[Z1x]=d[epsilon](0, [infinity]) and , then it is shown that a point process based on the...

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...