Extremal properties of M4 processes

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 $$\{{\mathbf {X}}_n=(X_{n1},\ldots ,X_{nd})\}_{n\ge 1}$$ { X n = ( X n 1 , … , X n d ) } n ≥ 1 having innovations $${\mathbf {Z}}_{l,n}=(Z_{l,n,1},\ldots ,Z_{l,n,d})$$ Z l , n = ( Z l , n , 1 , … , Z l , n , d ) with totally dependent margins for certain values of $$l,$$ l , $$l\in I_1,$$ l ∈ I 1 , and independent margins for the remaining values of $$l,$$ l , $$l\in I_2.$$ l ∈ I 2 . We obtain in this way a $$d$$ d -dimensional process $$\{{\mathbf {X}}_n\}_{n\ge 1}$$ { X n } n ≥ 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. Copyright Sociedad de Estadística e Investigación Operativa 2014