Spectral representations of sum- and max-stable processes

Abstract

To each max-stable process with α-Fréchet margins, α ∈ (0, 2), a symmetric α-stable process can be associated in a natural way. Using this correspondence, we deduce known and new results on spectral representations of max-stable processes from their α-stable counterparts. We investigate the connection between the ergodic properties of a stationary max-stable process and the recurrence properties of the non-singular flow generating its spectral representation. In particular, we show that a stationary max-stable process is ergodic iff the flow generating its spectral representation has vanishing positive recurrent component. We prove that a stationary max-stable process is ergodic (mixing) iff the associated SαS process is ergodic (mixing). We construct nonsingular flows generating the max-stable processes of Brown and Resnick.

Generalized Pickands constants and stationary max-stable processes

“…We denote by ε x the unit Dirac measure at x ∈ R. The Brown-Resnick process ξ W is both max-stable and stationary Kabluchko, 2009Kabluchko, , 2011Molchanov and Stucki, 2013;Molchanov et al, 2014). The stationarity means that the processes {ξ W (t), t ∈ R} and {ξ W (t + h), t ∈ R} have the same distribution for any h ∈ R. Moreover, the process ξ W arises naturally as the limit of suitably normalized pointwise maxima of independent copies of stationary Gaussian processes (Kabluchko et al, 2009, Theorem 17).…”

On the structure and representations of max-stable processes

“…[21], [22], [27], [29], and [30]). There are relatively few results of this nature about the structure of max-stable processes, with the notable exceptions of de Haan and Pickands [9], Davis and Resnick [6], and the very recent works of Kabluchko et al [16] and Kabluchko [14].…”

Simulation of Brown–Resnick processes

“…The notion of max-stable processes generated by non-singular flows has been introduced by de Haan and Pickands (1986); further results on the representations of max-stable processes have been obtained in Kabluchko (2009b) and Wang and Stoev (2009) by transferring some work of Rosiński (1995) on SαS-processes. Kabluchko et al (2009, Theorem 14) showed that a Brown-Resnick process is generated by a dissipative flow if Eq.…”

“…By Kabluchko (2009b) condition (1) holds only if Z (·) has a mixed moving maxima representation. In order to construct such a representation we repeat results from the proof of Theorem 14 in Kabluchko et al (2009).…”

“…This class is notable, as a subclass also occurs as the limit of maxima of independent copies of stationary and appropriately scaled Gaussian random fields (Kabluchko et al 2009) and, in a modified form, as the limit of empirical distribution functions (Kabluchko 2009a). Ergodic properties of the processes are studied by Kabluchko (2009b) and Wang and Stoev (2009) who discuss the decomposition into conservative and dissipative components; ergodicity and mixing properties are investigated by Kabluchko and Schlather (2009). Finally, Buishand et al (2008) and de Haan and Zhou (2008) used Brown-Resnick processes for modelling spatial rainfall.…”