Decomposition for multivariate extremal processes

The probability distribution of an extremal process in R(d) with independent max-increments is completely determined by its distribution function. The df of an extremal process is similar to the cdf of a random vector. It is a monotone function on (0,infinity) x R(d) with values in the interval [0, 1]. On the other hand the probability distribution of an extremal process is a probability measure on the space of sample functions. That is the space of all increasing right continuous functions y : (0, infinity) --> R(d) With the topology of weak convergence. A sequence of extremal processes converges in law if the probability distributions converge weakly. This is shown to be equivalent to weak convergence of the df's. An extremal process Y : [0,infinity) --> R(d) is generated by a point process on the space [0, infinity) x [-infinity, infinity)(d) and has a decomposition Y = X boolean OR Z as the maximum of two independent extremal processes with the same lower curve as the original process. The process X is the continuous part and Z contains the fixed discontinuities of the process Y. For a real valued extremal process the decomposition is unique; for a multivariate extremal process uniqueness breaks down, due to blotting.