On the spectral representation of symmetric stable processes

Hardin, Clyde D.

doi:10.1016/0047-259x(82)90073-2

Cited by 97 publications

(77 citation statements)

References 5 publications

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“…Our definition of a minimal representation is as in [26] (see also [9] and [13]). Other equivalent definitions have been investigated in [20] and [28].…”

Section: Minimal Representationsmentioning

confidence: 99%

“…At the same time, the structure and classification of sum-stable processes have been vigorously studied. Rosiński [26], building on the seminal works of Hardin [12], [13] about minimal representations, developed the important connection between stationary sum-stable processes and flows. This led to a number of important contributions on the structure of sum-stable processes (see, e.g.…”

Section: Introductionmentioning

confidence: 99%

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On the structure and representations of max-stable processes

Wang

Stoev

2010

Adv. Appl. Probab.

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We develop classification results for max-stable processes, based on their spectral representations. The structure of max-linear isometries and minimal spectral representations play important roles. We propose a general classification strategy for measurable maxstable processes based on the notion of co-spectral functions. In particular, we discuss the spectrally continuous-discrete, the conservative-dissipative, and the positive-null decompositions. For stationary max-stable processes, the latter two decompositions arise from connections to nonsingular flows and are closely related to the classification of stationary sum-stable processes. The interplay between the introduced decompositions of max-stable processes is further explored. As an example, the Brown-Resnick stationary processes, driven by fractional Brownian motions, are shown to be dissipative.

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“…Our definition of a minimal representation is as in [26] (see also [9] and [13]). Other equivalent definitions have been investigated in [20] and [28].…”

Section: Minimal Representationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the structure and representations of max-stable processes

Wang

Stoev

2010

Adv. Appl. Probab.

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“…, X(t n ). Nolan (1989) has relied on the L α -representations of symmetric α-stable random fields [see Hardin (1982) or Samorodnitsky and Taqqu (1994)] and the approximation properties of normed or quasi-normed linear spaces.…”

Section: Local Nondeterminism For Stable Processesmentioning

confidence: 99%

Properties of local-nondeterminism of Gaussian and stable random fields and their applications

Xiao¹

2009

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…For both classes, important role is played by the notion of spectral representation. Fundamental results on the existence and uniqueness of the spectral representation are due to Bretagnolle et al (1966), Schreiber (1972), Hardin (1982), Rosinski (1995) in the case of SαS processes and to de Haan (1984) and de Haan and Pickands (1986) in the case of maxstable processes.…”

Section: Introductionmentioning

confidence: 99%

Spectral representations of sum- and max-stable processes

Kabluchko

2009

Extremes

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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.

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On the spectral representation of symmetric stable processes

Cited by 97 publications

References 5 publications

On the structure and representations of max-stable processes

On the structure and representations of max-stable processes

Properties of local-nondeterminism of Gaussian and stable random fields and their applications

Spectral representations of sum- and max-stable processes

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