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 where is an iid sequence of random fields on [0,1]d with values in the Skorokhod space of càdlàg functions on [0,1]d equipped with the J1-topology. The coefficients [psi]i are deterministic real-valued fields on . The indices and t refer to the observation of the process at location and time t. For example, , could represent the time series of annual maxima of ozone levels at location . 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 in n years. By establishing a limit theory for point processes based on , 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 , 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 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.