Some Properties of Mixed Moving Maxima Processes on Z


  • Prof. Martin Schlather – University of Gottingen – Germany
  • FCUL (DEIO) – Campo Grande – Bloco C6 Piso 4 – Sala 6.4.30 – 14:00h
  • Quarta-feira, 17 de Novembro de 2010

Herglotz (1911) showed that any positive definite on Z can be represented by a Fourier integral. In a more abstract way, it has been shown that the set of correlation functions on Z is closed and convex, and its boundary is given by {CωЄ [-1; 1]Z : Cω(k) = exp(iωk); k Є Zω Є [-π; π)} (Sasvári, 1994).

The extremal correlation function (ecf) gives, roughly speaking, the conditional probability of an observed large value at lag h given one has observerd a large value right now. It is well-known that the ecf is a positive definite function, but the reverse does not hold in general.

We show that certain subsets of the ecf on Z are closed and convex, and we give the set of boundary functions in a limited number of cases. Tothis end we will consider set covariance functions on the real axis R that are closely related to the representation of max-stable processs by de Haan (1984). Finally, we will give some further properties of the ecf. The results help to reconstruct a max-stable process that fits the data.

This is joint work with Andree Ehlert.


L. de Haan. A spectral representation for max-stable processes. Ann. Probab.,12:1194{1204, 1984.G. Herglotz. Uber Potenzreihen mit positivem, reellen Teil im Einheitskreis.Leipziger Berichte, math.-phys. Kl., 63:501{511, 1911.Z. Sasv_ari. Positive Definite and Definitizable Functions. Akademie Verlag,Berlin, 1994.