Summary
Let (ξ k ) ∞ k =−∞ be a stationary sequence of random variables, and, forA⊂ℝ, let\(M_n (A): = \mathop V\limits_{k/n \in A} \gamma _n (\xi _k )\) where γ n is an affine transformation of ℝ (has the forma n·+b n,a n>0,b n∈ℝ). ThenM n is a random sup measure, that is,\(M_n (\mathop U\limits_\alpha G_\alpha ) = \mathop V\limits_\alpha M_n (G_\alpha )\) for arbitrary collections of open setsG α. We show that the possible limiting random sup measures for such sequences (M n) are those which are stationary (M(·+b)= d M forb∈ℝ) and self-similar (M(a·)= d δloga(M) fora>0, where δ is an affine transformation of ℝ). By applying simple transformations, we need only study stationaryM such thatM(a·)= d aM fora>0. We show that these processes retain some but not all of the properties of the classical case. In particular, we display a nontrivial example such thatt↦M (0,t] is continuous wp1. The classical planar point process representation of extremal processes is a special case of the present approach, but is not adequate for describing all possible limits.
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O'Brien, G.L., Torfs, P.J.J.F. & Vervaat, W. Stationary self-similar extremal processes. Probab. Th. Rel. Fields 87, 97–119 (1990). https://doi.org/10.1007/BF01217748
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DOI: https://doi.org/10.1007/BF01217748