, Volume 23, Issue 2, pp 388–408 | Cite as

Extremal properties of M4 processes

  • A. P. Martins
  • H. Ferreira
Original Paper


The existence of data with different dependence structures motivates the development of models which can capture several types of dependence. In this paper we consider a stationary sequence of moving maxima vectors \(\{{\mathbf {X}}_n=(X_{n1},\ldots ,X_{nd})\}_{n\ge 1}\) having innovations \({\mathbf {Z}}_{l,n}=(Z_{l,n,1},\ldots ,Z_{l,n,d})\) with totally dependent margins for certain values of \(l,\) \(l\in I_1,\) and independent margins for the remaining values of \(l,\) \(l\in I_2.\) We obtain in this way a \(d\)-dimensional process \(\{{\mathbf {X}}_n\}_{n\ge 1}\) whose extremal dependence, measured by the tail dependence coefficients, lies between asymptotic independence and total dependence. The extremal properties of these M4 processes are studied and examined both theoretically and through simulation studies: we derive the multivariate extremal index, the tail dependence coefficients and co-movement indices.


Multivariate extremes M4 processes Tail dependence  Extremal index Co-movement index 

Mathematics Subject Classification (2000)




The authors are grateful to the reviewers for their insightful comments and suggestions. This research was partially supported by the research unit “Centro de Matemática” of the University of Beira Interior and the research project PEst-OE/MAT/UI0212/2014 through the Foundation for Science and Technology (FCT) co-financed by FEDER/COMPETE.


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Copyright information

© Sociedad de Estadística e Investigación Operativa 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Beira InteriorCovilhãPortugal

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