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

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• 作者：Richard A. Davis ; Thomas Mikosch
• 关键词：Regular variation ; Extremes ; Max-stable processes ; Space-time processes
• 刊名：Stochastic Processes and their Applications
• 出版年：2008
• 出版时间：April 2008
• 年：2008
• 卷：118
• 期：4
• 页码：560-584
• 全文大小：1121 K

文摘

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 41a17ffd81fe9753a60cad51274bb"" title=""Click to view the MathML source"">[0,1]^d equipped with the 1a2aeb0ed09fc50a2026114876f977"" title=""Click to view the MathML source"">J₁-topology. The coefficients ψ_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.