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

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 Xt(s)=∑i=0∞ψi(s)Zt−i(s),s∈[0,1]d, where (Zt)t∈Z(Zt)t∈Z is an iid sequence of random fields on [0,1]d[0,1]d with values in the Skorokhod space D([0,1]d)D([0,1]d) of càdlàg functions on [0,1]d[0,1]d equipped with the J1J1-topology. The coefficients ψiψi are deterministic real-valued fields on D([0,1]d)D([0,1]d). The indices s and tt refer to the observation of the process at location s and time tt. For example, Xt(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[0,1]2 does not exceed a given standard level f∈D([0,1]2)f∈D([0,1]2) in nn years. By establishing a limit theory for point processes based on (Xt(s)), t=1,…,nt=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]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)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.