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 tail process is equivalent to multivariate regular variation of finite cuts of the original process. Certain remarkable properties of the tail process are exploited to shed new light on known results on certain point processes of extremes. The theory is shown to be applicable with great ease to stationary solutions of stochastic autoregressive processes with random coefficient matrices, an interesting special case being a recently proposed factor GARCH model. In this class of models, the distribution of the tail process is calculated by a combination of analytical methods and a novel sampling algorithm.

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 dependence structure between neighboring observations when both are large. Regular variation also plays a vital role in establishing the large sample behavior of a variety of statistics from a GARCH process including the sample mean and the sample autocovariance and autocorrelation functions. In particular, if the 4th moment of the process does not exist, the rate of convergence of the sample autocorrelations becomes extremely slow, and if the 2nd moment does not exist, the sample autocorrelations have non-degenerate limit distributions.

We establish the equivalence between the multivariate regular variation of a random vector and the univariate regular variation of all linear combinations of the components of such a vector. According to a classical result of Kesten [Acta Math. 131 (1973) 207-248], this result implies that stationary solutions to multivariate linear stochastic recurrence equations are regularly varying. Since GARCH processes can be embedded in such recurrence equations their finite-dimensional distributions are regularly varying. Introduction.Much of the early development of regular variation in the multivariate setting had its genesis in extreme value theory. There is a natural connection between limit theory of component maxima of iid random vectors and multivariate regular variation. Excluding some special degenerate cases, a random vector with positive components is in the maximum domain of attraction of a multivariate extreme value distribution with the same Fréchet marginals if and only if the vector has a distribution which is regularly varying; for details see, for example, [17], [18], Chapter 5. Early on, multivariate regular variation has been used in the theory of summation of iid random vectors to characterize the domains of attraction of stable distributions; see [20]. More recently, multivariate regular variation has been found to be a key concept in problems that go beyond extreme value theory. In particular, it has been used to describe the weak limits of point processes constructed from stationary sequences of random vectors; see [6,7]. These weak convergence results provided the key ingredients for deriving the weak limiting behavior of sample autocovariances and sample autocorrelations of stationary sequences of random variables with regularly varying tails. In fact, one can apply a continuous mapping argument to obtain the weak convergence of these and other sum-type functionals of a stationary sequence from the weak convergence of appropriately chosen point processes, where the joint distributions of the points constituting the process satisfy a multivariate regular variation condition.

Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable Lévy process. The main result in the paper is that for a stationary, regularly varying sequence for which clusters of high-threshold excesses can be broken down into asymptotically independent blocks, the properly centered partial sum process still converges to a stable Lévy process. Due to clustering, the Lévy triple of the limit process can be different from the one in the independent case. The convergence takes place in the space of càdlàg functions endowed with Skorohod's M1 topology, the more usual J1 topology being inappropriate as the partial sum processes may exhibit rapid successions of jumps within temporal clusters of large values, collapsing in the limit to a single jump. The result rests on a new limit theorem for point processes which is of independent interest. The theory is applied to moving average processes, squared GARCH(1, 1) processes and stochastic volatility models. . This reprint differs from the original in pagination and typographic detail. 1 2 B. BASRAK, D. KRIZMANIĆ AND J. SEGERS tributional behavior of the D[0, 1] valued process V n (t) = a −1 n (S ⌊nt⌋ − ⌊nt⌋b n ), t ∈ [0, 1],

We show a new functional limit theorem for weakly dependent regularly varying sequences of random vectors. As it turns out, the convergence takes place in the space of R d valued càdlàg functions endowed with the so-called weak M 1 topology. The theory is illustrated on two examples. In particular, we demonstrate why such an extension of Skorohod's M 1 topology is actually necessary for the limit theorem to hold.