Gaussian Random Functions

Lifshits, Mikhail

doi:10.1007/978-94-015-8474-6

Cited by 318 publications

(214 citation statements)

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“…This computation allows us, in particular, to assert that Dudley integral is finite on Θ. It yields Lifshits (1995) The last inequality shows also that there exist τ > 0 such that for any s ∈ S, satisfying lim sup a→∞ a τ s −1 (a) < ∞, Assumption 3 is fulfilled.…”

Section: Proof Of Theoremmentioning

confidence: 82%

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Upper functions for positive random functionals. I. General setting and Gaussian random functions

Lepski

2013

Math. Meth. Stat.

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In this paper we are interested in finding upper functions for a collection realvalued random variables Ψ χ θ , θ ∈ Θ . Here {χ θ , θ ∈ Θ} is a family of continuous random mappings, Ψ is a given sub-additive positive functional and Θ is a totally bounded subset of a metric space. We seek a non-random function U : Θ → R+ such that sup θ∈Θ Ψ χ θ −U (θ) + is "small" with prescribed probability. We apply the results obtained in the general setting to the variety of problems related to gaussian random functions and empirical processes.

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Section: Proof Of Theoremmentioning

confidence: 82%

“…In view of the latter remark we can consider the set Θ those entropy obeys the restriction which is closer to the minimal one (c.f. Sudakov lower bound for gaussian random functions Lifshits (1995)). We note, however, that there exist examples where s has to be chosen on a more special way (see Theorem 1).…”

Section: Discussionmentioning

confidence: 99%

Upper functions for positive random functionals. I. General setting and Gaussian random functions

Lepski

2013

Math. Meth. Stat.

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“…Using the continuous functional F (f ) = f ∞ and (26), we obtain: It follows from (27) that µ(L) = 1. By (Lifshits, 1995, Section 9, Proposition 1) H ξ ⊂ L. From Lemma 2 we obtain the following Bernsteintype theorem from approximation theory:…”

Section: 2mentioning

confidence: 88%

Functional Limit Theorems for Multiparameter Fractional Brownian Motion

Malyarenko

2006

J Theor Probab

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Abstract. We prove a general functional limit theorem for multiparameter fractional Brownian motion. The functional law of the iterated logarithm, functional Lévy's modulus of continuity and many other results are its particular cases. Applications to approximation theory are discussed. IntroductionLet B(t) = B(t, ω), t ≥ 0, ω ∈ Ω be the Brownian motion on the probability space (Ω, F, P). The law of the iterated logarithm (Khintchin, 1924) states that P ω : lim sup t→∞ B(t, ω) √ 2t log log t = 1 = 1.We abbreviate this as(1) lim sup t→∞ B(t) √ 2t log log t = 1 P − a. s., where a. s. stands for almost surely. The functional counterpart to the law of the iterated logarithm was discovered by (Strassen, 1964). Let C[0, 1] be the Banach space of all continuous functions f : [0, 1] → R with the uniform topology generated by the maximum norm f ∞ = max t∈[0,1] |f (t)|.

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“…A standard Gaussian field on E is a collection of random variables {W x ; x ∈ E} any linear combination of which is Gaussian and such that EW x = 0 and EW 2 x = 1 for x ∈ E (see [11] or [12] for more details on Gaussian fields) . The covariance function of W Σ x,y = EW x W y , x, y ∈ E is semi-definite positive.…”

Section: Gaussian Level Sets Covariancesmentioning

confidence: 99%

Realisability conditions for second-order marginals of biphased media

Lachièze-Rey

2014

Random Struct. Alg.

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This paper concerns the second order marginals of biphased random media. We give discriminating necessary conditions for a bivariate function to be such a valid marginal, and illustrate our study with two practical applications: (1) the spherical variograms are valid indicator variograms if and only if they are multiplied by a sufficiently small constant, which upper bound is estimated, and (2) not every covariance/indicator variogram can be obtained with a Gaussian level set. The theoretical results backing this study are contained in a companion paper.

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Gaussian Random Functions

Cited by 318 publications

References 0 publications

Upper functions for positive random functionals. I. General setting and Gaussian random functions

Upper functions for positive random functionals. I. General setting and Gaussian random functions

Functional Limit Theorems for Multiparameter Fractional Brownian Motion

Realisability conditions for second-order marginals of biphased media

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