Semi-self-similar extremal processes

Abstract

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 = (τα, Lα): α > 0}, i.e. g ∘ ηα = g ∀ α > 0, then g is self-similar. If g is invariant w.r.t. the cyclic group {η∘(n), n ∈Z} of a time-space change ν, then g is semi-self-similar. The semi-self-similar extremal processes are limiting for sequences of extremal processes Yn(t)=Ln−1 ∘ Y ∘ τn (t) if going along a geometrically increasing subsequence kn ∼ ϕn, 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.