Continuous Breuer-Major theorem: tightness and non-stationarity

Campese, Simon; Nourdin, Ivan; Nualart, David

doi:10.48550/arxiv.1807.09740

Cited by 3 publications

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“…Due to results by Nualart and Peccati [17] and Peccati and Tudor [18] (or see Theorem 2.1 of [5]), the convergence of the finite dimensional distributions of Z s to those of the Brownian motion √ V B y follows if we show that the covariances of the corresponding projections on each Wiener chaos converge. Namely for any q ≥ d and y 1 , y 2 > 0,…”

Section: 3mentioning

confidence: 95%

“…This remains an unaddressed question in the literature in the case of vectors. The approach here is similar to the method that has been employed in [5] and [14], namely using the representation by means of the Malliavin divergence operator, which is obtained through a shift operator, and applying Meyer inequalities to show tightness. However, in the case of vectors fields, this approach is more involved and requires the introduction of weighted shift operators.…”

Section: Introductionmentioning

confidence: 99%

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Continuous Breuer-Major theorem for vector valued fields

Nualart

Tilva

2020

Stochastic Analysis and Applications

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Let ξ : Ω × R n → R be zero mean, mean-square continuous, stationary, Gaussian random field with covariance function r(x) = E[ξ(0)ξ(x)] and let G : R → R such that G is square integrable with respect to the standard Gaussian measure and is of Hermite rank d. The Breuer-Major theorem in it's continuous setting gives that, if r ∈ L d (R n ), then the finite dimensional distributions of G(ξ(x))] dx converge to that of a scaled Brownian motion as s → ∞. Here we give a proof for the case when ξ : Ω × R n → R m is a random vector field. We also give a proof for the functional convergence in C([0, ∞)) of Zs to hold under the condition that for some p > 2, G ∈ L p (R m , γm) where γm denotes the standard Gaussian measure on R m and we derive expressions for the asymptotic variance of the second chaos component in the Wiener chaos expansion of Zs(1).2010 Mathematics Subject Classification. 60F05, 60F17, 60G15, 60G60, 60H07.

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Section: 3mentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Continuous Breuer-Major theorem for vector valued fields

Nualart

Tilva

2020

Stochastic Analysis and Applications

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“…We first assemble the convergence of scalar processes, extending them to the same larger class of functions. To extend the convergence to the Hölder topology, we follow [CNN18] and use Malliavin calculus to obtain moment bounds. This section is quite short, only about 3 pages.…”

Section: Theorem B [Clt]mentioning

confidence: 99%

“…In the lemma below we estimate the moments of t 0 G(x r ε )dr, where we need the multiple Wiener-Itô-integral representation above to transfer the correlation function to L 2 norms of indicator functions. We use an idea from [CNN18] for the estimates below.…”

Section: Moment Boundsmentioning

confidence: 99%

Homogenization with fractional random fields

Gehringer,

2019

Preprint

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We consider a system of differential equations in a fast long range dependent random environment and prove a homogenization theorem involving multiple scaling constants. The effective dynamics solves a rough differential equation, which is 'equivalent' to a stochastic equation driven by mixed Itô integrals and Young integrals with respect to Wiener processes and Hermite processes. Lacking other tools we use the rough path theory for proving the convergence, our main technical endeavour is on obtaining an enhanced scaling limit theorem for path integrals (Functional CLT and non-CLT's) in a strong topology, the rough path topology, which is given by a Hölder distance for stochastic processes and their lifts. In dimension one we also include the negatively correlated case, for the second order / kinetic fractional BM model we also bound the error.

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“…In particular, in the case where the limiting random variable is a Gaussian process, our results yield a universal and quantitative method to prove functional central limit theorems, and can be regarded as an alternative to the (non-quantitative) strategy of proving convergence of finite-dimensional distributions and tightness of the approximating sequence. To illustrate this method, we quantify a functional version of the celebrated Breuer-Major theorem (recently proved in [NN18] (see also [CNN18,HN18] for related results), thus assessing the speed of convergence of stochastic processes (U t ) t≥0 of the form…”

Section: Introductionmentioning

confidence: 99%

Approximation of Hilbert-valued Gaussians on Dirichlet structures

Bourguin¹,

Campese²

2019

Preprint

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We introduce a framework to derive quantitative central limit theorems for the approximation of arbitrary non-degenerate Gaussian random variables taking values in a separable Hilbert space. In particular, our method provides an alternative to the usual (non-quantitative) finite dimensional distribution convergence and tightness argument for proving functional convergence of stochastic processes. We also derive four moments theorems for Hilbert-valued random variables with possibly infinite chaos expansion, which also include, as special cases, all finite-dimensional four moments theorems for Gaussian approximation in a diffusive context proved earlier by various authors. Our main ingredients are infinite-dimensional versions of Stein's method as introduced by Shih, and the so-called Gamma calculus. As an application, we derive rates of convergence in the functional Breuer-Major theorem recently proved by Nourdin and Nualart.

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Continuous Breuer-Major theorem: tightness and non-stationarity

Cited by 3 publications

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Continuous Breuer-Major theorem for vector valued fields

Continuous Breuer-Major theorem for vector valued fields

Homogenization with fractional random fields

Approximation of Hilbert-valued Gaussians on Dirichlet structures

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