Extreme value theory for space–time processes with heavy-tailed distributions

Abstract

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, insurance, telecommunications, meteorology, and hydrology. Regular variation provides a convenient and unified background for studying multivariate extremes when heavy tails are present. In this paper, we study the extreme value behavior of the space–time process given by X t ( s ) = ∑ i = 0 ∞ ψ i ( s ) Z t − i ( s ) , s ∈ [ 0 , 1 ] d , where ( Z t ) t ∈ Z is an iid sequence of random fields on [ 0 , 1 ] d with values in the Skorokhod space D ( [ 0 , 1 ] d ) of càdlàg functions on [ 0 , 1 ] d equipped with the J 1 -topology. The coefficients ψ i are deterministic real-valued fields on D ( [ 0 , 1 ] d ) . The indices s and t refer to the observation of the process at location s and time t . For example, X t ( s ) , t = 1 , 2 , … , could represent the time series of annual maxima of ozone levels at location s . The problem of interest is determining the probability that the maximum ozone level over the entire region [ 0 , 1 ] 2 does not exceed a given standard level f ∈ D ( [ 0 , 1 ] 2 ) in n years. By establishing a limit theory for point processes based on ( X t ( s ) ) , t = 1 , … , n , we are able to provide approximations for probabilities of extremal events. This theory builds on earlier results of de Haan and Lin [L. de Haan, T. Lin, On convergence toward an extreme value distribution in C [ 0 , 1 ] , Ann. Probab. 29 (2001) 467–483] and Hult and Lindskog [H. Hult, F. Lindskog, Extremal behavior of regularly varying stochastic processes, Stochastic Process. Appl. 115 (2) (2005) 249–274] for regular variation on D ( [ 0 , 1 ] d ) and Davis and Resnick [R.A. Davis, S.I. Resnick, Limit theory for moving averages of random variables with regularly varying tail probabilities, Ann. Probab. 13 (1985) 179–195] for extremes of linear processes with heavy-tailed noise.