Spectral representations of sum- and max-stable processes

By Zakhar Kabluchko. 2009

78 citations recorded by the source corpus.

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 non-singular flows generating the max-stable processes of Brown and Resnick.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/18756faeb51b06688a8b20accca953df4338ccdf

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

1. Richard A. Davis
2. Martin Schlather
3. Thomas Mikosch
4. Zakhar Kabluchko
5. Stilian A. Stoev
6. Yizao Wang
7. Yuwei Zhao
8. Marco Oesting

In Statistical and Nonlinear Physics

1. Martin Schlather
2. Thomas Mikosch
3. Yuwei Zhao
4. Richard A. Davis
5. Stilian A. Stoev
6. Yizao Wang

In Statistics and Probability

1. Martin Schlather
2. Richard A. Davis
3. Thomas Mikosch
4. Zakhar Kabluchko
5. Yuwei Zhao
6. Yizao Wang
7. Stilian A. Stoev
8. Mathieu Ribatet

Citations in the corpus

• Measures of serial extremal dependence and their estimation (2013, background): “Since Gumbel or Weibull distributed random variables can be obtained by suitable increasing transformations of a Fréchet random variable any result for max-stable processes with Fréchet marginals can be formulated in terms of the transformed processes with Gumbel or Weibull marginals; see for exam”
• On the structure and representations of max-stable processes (2009, background): “The combination of the two approaches provides a clearer picture on the structure of max- and sum-stable processes as well as their interplay.”
• Simulation of Brown–Resnick processes (2012, background): “By Kabluchko (2009b) condition (1) holds only if Z(·) has a mixed moving maxima representation.”
• Ergodic properties of max-infinitely divisible processes (2009, methodology): “Later, the result of [26] was used in [8] to show that a max–stable process is ergodic iff the flow generating its spectral representation has no positive recurrent component.”