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

Xiao, Yimin

doi:10.5802/afst.1117

Cited by 67 publications

(85 citation statements)

References 64 publications

(59 reference statements)

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“…The paper continues the research initiated in the paper [4] concerning the real harmonizable multifractional process and those of the papers [15,16] concerning the fractional Brownian and stable fields.…”

Section: Introductionmentioning

confidence: 54%

“…is an increment of a harmonizable (non-multi) fractional stable field considered in [15]. Theorem 3.5 in [15] claims that…”

Section: Georgiy Shevchenkomentioning

confidence: 99%

“…The constants in the latter inequality depend on H but it is easily seen from the proof of this fact in [15] that the constants are bounded away from zero under the assumption that the Hurst parameters are separated from 0 and 1. Thus we can assume that these constants are universal.…”

Section: Georgiy Shevchenkomentioning

confidence: 99%

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Local properties of a multifractional stable field

Shevchenko

2013

Theor. Probability and Math. Statist.

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Section: Introductionmentioning

confidence: 54%

“…is an increment of a harmonizable (non-multi) fractional stable field considered in [15]. Theorem 3.5 in [15] claims that…”

Section: Georgiy Shevchenkomentioning

confidence: 99%

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Local properties of a multifractional stable field

Shevchenko

2013

Theor. Probability and Math. Statist.

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“…The notion of strong local nondeterminism was later developed to investigate the regularity of local times, small ball probabilities and other sample path properties of Gaussian processes and Gaussian random fields. We refer to Xiao (2006Xiao ( , 2007 for more information on the history and applications of the properties of local nondeterminism.…”

Section: Properties Of Strong Local Nondeterminismmentioning

confidence: 99%

Sample Path Properties of Anisotropic Gaussian Random Fields

Xiao¹

Lecture Notes in Mathematics

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141

227

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Anisotropic Gaussian random fields arise in probability theory and in various applications. Typical examples are fractional Brownian sheets, operator-scaling Gaussian fields with stationary increments, and the solution to the stochastic heat equation.This paper is concerned with sample path properties of anisotropic Gaussian random fields in general. Let X = {X(t), t ∈ R N } be a Gaussian random field with values in R d and with parameters H 1 , . . . , H N . Our goal is to characterize the anisotropic nature of X in terms of its parameters explicitly.Under some general conditions, we establish results on the modulus of continuity, small ball probabilities, fractal dimensions, hitting probabilities and local times of anisotropic Gaussian random fields. An important tool for our study is the various forms of strong local nondeterminism.

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“…. , x ω (t q ) is comparable with the variance of x ω (t 1 ) − x ω (t j ) for the j for which |t 1 − t j | is least), see Be,Xi06,Xi11 [3,25,26]. It may be shown that the calculations are essentially unaffected if, for a suitably large m, we consider the numbers in [0, 1] to base m and identify the base m number 0.a 1 a 2 a 3 .…”

Section: Images Of Measures Under Gaussian Processesmentioning

confidence: 99%