Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions

Eden, Richard; Víquez, Juan

doi:10.1016/j.spa.2014.09.001

Cited by 22 publications

(30 citation statements)

References 23 publications

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“…There appear two terms R[10, i] (i = 1, 2) by (7.28) for F = G • R [4]. There appear two terms R[4, i] (i = 1, 2) by (7.28) for F = G • R [3]. This is similar to the case R [4].…”

Section: Asymptotic Expansion For Measurable Functionsmentioning

confidence: 60%

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Asymptotic expansion of Skorohod integrals

Nualart¹,

Yoshida²

2019

Electron. J. Probab.

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Asymptotic expansion of the distribution of a perturbation Z n of a Skorohod integral jointly with a reference variable X n is derived. We introduce a second-order interpolation formula in frequency domain to expand a characteristic functional and combine it with the scheme developed in the martingale expansion. The second-order interpolation and Fourier inversion give asymptotic expansion of the expectation E[f (Z n , X n )] for differentiable functions f and also measurable functions f . In the latter case, the interpolation method connects the two non-degeneracies of variables for finite n and ∞. Random symbols are used for expressing the asymptotic expansion formula. Quasi tangent, quasi torsion and modified quasi torsion are introduced in this paper. We identify these random symbols for a certain quadratic form of a fractional Brownian motion and for a quadratic from of a fractional Brownian motion with random weights. For a quadratic form of a Brownian motion with random weights, we observe that our formula reproduces the formula originally obtained by the martingale expansion.

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“…There appear two terms R[10, i] (i = 1, 2) by (7.28) for F = G • R [4]. There appear two terms R[4, i] (i = 1, 2) by (7.28) for F = G • R [3]. This is similar to the case R [4].…”

Section: Asymptotic Expansion For Measurable Functionsmentioning

confidence: 60%

“…Then, integrating ∂ θ ϕ n (θ; ψ n ) and using the expression for R (3) n (z, x) and the decomposition (2.3), yields…”

Section: Perturbation Of a Skorohod Integralmentioning

confidence: 99%

Asymptotic expansion of Skorohod integrals

Nualart¹,

Yoshida²

2019

Electron. J. Probab.

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“…The following result, a proof of which can be found in [EVq15], combines results from [KT12] and [EVq15] and provides sufficient conditions under which useful estimates for f ′ h can be obtained. Lemma 2.4.…”

Section: Stein's Methods For Invariant Measures Of Diffusionsmentioning

confidence: 81%

“…Based on the above Stein lemma, one can use the now well established Stein methodology to quantitatively measure the distance between the law of a random variable X and the law of a random variable Z corresponding to an invariant measure of a diffusion. The generalization of the original Stein method to invariant measures of diffusions has been recently studied in [KT12] and further developed in [EVq15]. In order to present this method, we need to introduce separating classes of functions and probabilistic distances.…”

Section: Stein's Methods For Invariant Measures Of Diffusionsmentioning

confidence: 99%

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Four moments theorems on Markov chaos

Bourguin¹,

Campese²,

Leonenko³

et al. 2019

Ann. Probab.

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A. We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions. For elements of a Markov chaos, this bound can be reduced to just the first four moments.2010 Mathematics Subject Classification. 60F05, 60J35, 60J99.

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Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

Gaunt

2018

J Theor Probab

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We use Stein's method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren [37] for Stein's method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and it first two derivatives, of the Rayleigh Stein equation.where the test function h is real-valued. The second step is to solve (1.2) for f h and obtain suitable bounds for the solution. Finally, to approximate the distribution of a random variable of interest W by the target distribution q, one may evaluate both sides of (1.2) at W , take expectations and finally take the supremum of both sides over a class of functions H to obtain d H (W, Y ) := sup h∈H |E[h(W )] − E[h(Y )]| = sup h∈H E[Af h (W )].

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Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions

Cited by 22 publications

References 23 publications

Asymptotic expansion of Skorohod integrals

Asymptotic expansion of Skorohod integrals

Four moments theorems on Markov chaos

Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

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