Semi-self-similar extremal processes

Let g be the distribution function (d.f.) of an extremal process Y. If g is invariant with respect to a continuous one-parameter group of time-space changes {ηα= (rα, Lα): α > 0}, i.e., g o ηα = g ∀α > 0, then g is self-similar. If g is invariant w.r.t. the cyclic group {ηo(η), η ∈ Z} of a time-space change η, then g is semi-self-similar. The semi-self-similar extremal processes are limiting for sequences of extremal processes Yη(t) = Lη -1 o Y o τη(t) if going along a geometrically increasing subsequence kη ∼ φη, φ > 1, n → ∞. The main properties of multivariate semi-self-similar extremal processes and some examples are discussed in the paper. The results presented are an analog of the theory of semi-self-similar processes with additive increments developed by Maejima and Sato in 1997. © 2000 Kluwer Academic/Plenum Publishers.