Approximation du temps local des surfaces gaussiennes

Berzin, Corinne; Wschebor, Mario

doi:10.1007/bf01195880

Cited by 7 publications

(11 citation statements)

References 9 publications

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“…Now let us enunciate a lemma for which a proof can be founded for example in Berzin and Wschebor [11…”

Section: Convergence In Law For the Estimatorsmentioning

confidence: 99%

Estimation of local anisotropy based on level sets

Berzin

2021

Electron. J. Probab.

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Consider an affine field X : R 2 → R, that is a process equal in law to Z(A.t), where Z is isotropic and A : R 2 → R 2 is a linear self-adjoint transformation. The field X and transformation A will be supposed to be respectively Gaussian and definite positive. Denote 0 < λ := λ 2 λ 1 1 the ratio of the eigenvalues of A, let λ1, λ2 with λ2 λ1. This paper is aimed at testing the null hypothesis "X is isotropic" versus the alternative "X is affine". Roughly speaking, this amounts to testing "λ = 1" versus "λ < 1". By setting level u in R, this is implemented by the partial observations of process X through some particular level functionals viewed over a square Tn, which grows to R 2 . This leads us to provide estimators for the affinity parameters that are shown to be almost surely consistent. Their asymptotic normality results provide confidence intervals for parameters.This paper offered an important opportunity to study general level functionals near the level u and for a fixed bounded rectangle T of R 2 , part of the difficulties arises from the fact that the topology of level set CT,X (u) = {t ∈ T : X(t) = u} can be irregular, even if the trajectories of X are regular. A significant part of the paper is dedicated to show the L 2 -continuity in the level u of these general functionals.

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“…Now let us enunciate a lemma for which a proof can be founded for example in Berzin and Wschebor [11…”

Section: Convergence In Law For the Estimatorsmentioning

confidence: 99%

Estimation of local anisotropy based on level sets

Berzin

2021

Electron. J. Probab.

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“…We will establish a theorem that gives assumptions on X and Y such that these formulas will hold for all y in R j . Specifically, we will assume continuity of both members of ( 18) and (19), restricting ourselves to the set D r X and proving the equality for any y fixed in R j . Before going any further, let us state two assumptions that are useful for what follows.…”

Section: Rice Formula For a Regular Level Setmentioning

confidence: 99%

“…The aim of this section is to approximate the local time of X over T at a given level u, say L X (u, T ), by the length of curves of the u level set of process X ε , say σ d−1 (C T,Xε (u)). Some of the results that will be presented in Sections 4.6.1 and 4.6.2 have been partially discussed in [19]. We can cite some references of papers that have dealt on the same subject in the case where d = 1.…”

Section: Local Time and Length Of Curves Of Level Setmentioning

confidence: 99%

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Kac-Rice formula: A contemporary overview of the main results and applications

Berzin¹,

Latour²,

León³

2022

Preprint

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“…(For a proof in the case of the length of the level curve, see Kratz and León (2001). The condition on the derivatives is given in Berzin and Wschebor (1993). Our case could be treated in a similar fashion.)…”

Section: Chaos Expansion and Central Limit Theoremmentioning

confidence: 99%