Abstract
The probability distribution of an extremal process in Rd 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, ∞) × Rd 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, ∞) → Rd 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, ∞) → Rd is generated by a point process on the space [0, ∞) × [-∞, ∞)d and has a decomposition Y = X v 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.