Extremal tt processes: Elliptical domain of attraction and a spectral representation

The extremal t process was proposed in the literature for modeling spatial extremes within a copula framework based on the extreme value limit of elliptical t distributions (Davison et al. (2012) [5]). A major drawback of this max-stable model was the lack of a spectral representation such that for instance direct simulation was infeasible. The main contribution of this note is to propose such a spectral construction for the extremal t process. Interestingly, the extremal Gaussian process introduced by Schlather (2002) [22] appears as a special case. We further highlight the role of the extremal t process as the maximum attractor for processes with finite-dimensional elliptical distributions. All results naturally also hold within the multivariate domain.

[1]  A. McNeil,et al.  The t Copula and Related Copulas , 2005 .

[2]  Gennady Samorodnitsky,et al.  Subexponentiality of the product of independent random variables , 1994 .

[3]  C. Klüppelberg,et al.  Semi‐Parametric Models for the Multivariate Tail Dependence Function – the Asymptotically Dependent Case , 2008 .

[4]  L. Breiman,et al.  On Some Limit Theorems Similar to the Arc-Sin Law , 1965 .

[5]  Conditional simulation of extremal Gaussian processes , 2012, 1202.4737.

[6]  Anthony C. Davison,et al.  Statistical Modelling of Spatial Extremes , 2012 .

[7]  J. Hüsler.  Maxima of normal random vectors: between independence and complete dependence , 1989 .

[8]  Mathieu Ribatet,et al.  Conditional simulations of Brown-Resnick processes , 2011 .

[9]  Mathieu Ribatet,et al.  Conditional simulation of max-stable processes , 2012, 1208.5376.

[10]  Sidney I. Resnick,et al.  Extreme values of independent stochastic processes , 1977 .

[11]  A. Davison,et al.  Statistical Modeling of Spatial Extremes , 2012, 1208.3378.

[12]  Jonathan A. Tawn,et al.  A dependence measure for multivariate and spatial extreme values: Properties and inference , 2003 .

[13]  Laurens de Haan,et al.  Stationary max-stable fields associated to negative definite functions. , 2008, 0806.2780.

[14]  Henning Omre,et al.  T-distributed Random Fields : A Parametric Model for Heavy-tailed Random Fields , 2006 .

[15]  E. Hashorva.  On the regular variation of elliptical random vectors , 2006 .

[16]  L. de Haan,et al.  A Spectral Representation for Max-stable Processes , 1984 .

[17]  F. Lindskog,et al.  Multivariate extremes, aggregation and dependence in elliptical distributions , 2002, Advances in Applied Probability.

[18]  J. Segers.  MAX-STABLE MODELS FOR MULTIVARIATE EXTREMES , 2012, 1204.0332.

[19]  Elliptical triangular arrays in the max-domain of attraction of Hüsler¿Reiss distribution , 2005 .

[20]  L. Haan,et al.  Extreme value theory : an introduction , 2006 .

[21]  J. Teugels,et al.  Statistics of Extremes , 2004 .

[22]  Richard A. Davis,et al.  Extreme value theory for space-time processes with heavy-tailed distributions , 2008 .

[23]  J. Hüsler,et al.  Laws of Small Numbers: Extremes and Rare Events , 1994 .

[24]  Martin Schlather,et al.  Models for Stationary Max-Stable Random Fields , 2002 .

[25]  Aristidis K. Nikoloulopoulos,et al.  Extreme value properties of multivariate t copulas , 2009 .

[26]  C. Klüppelberg,et al.  Estimating the tail dependence function of an elliptical distribution , 2007 .

[27]  Kai-Tai Fang,et al.  On the Theory of Multivariate Elliptically Contoured Distributions and Their Applications. , 1982 .

[28]  C. Dombry,et al.  Regular conditional distributions of max infinitely divisible processes , 2011, 1109.6492.

[29]  Extremes of independent chi-square random vectors , 2012 .

[30]  Anthony C. Davison,et al.  Geostatistics of extremes , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.