Multivariate Gaussian approximations on Markov chaoses

Campese, Simon; Nourdin, Ivan; Peccati, Giovanni; Poly, Guillaume

doi:10.48550/arxiv.1510.02105

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Fourth Moment Theorems For Complex Gaussian Approximation1

Introduction1

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2015

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“…We start by introducing the notion of chaos, which for the real case was given in [ACP14] and extended to the multidimensional case in [CNPP15]. These definitions can be generalized as follows to the complex setting.…”

Section: Fourth Moment Theorems For Complex Gaussian Approximationmentioning

confidence: 99%

“…Of course, from a purely algebraic point of view, our approach is equivalent to the multidimensional real case, which has been treated in [CNPP15], and indeed one could handle sequences of complex valued random variables by separating real and imaginary part and considering them as two-dimensional real vectors. For the complex Ornstein-Uhlenbeck generator, such an ad-hoc strategy has been followed in [CL14] and [Che14].…”

Section: Introductionmentioning

confidence: 99%

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Fourth Moment Theorems for complex Gaussian approximation

Campese

2015

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We prove a bound for the Wasserstein distance between vectors of smooth complex random variables and complex Gaussians in the framework of complex Markov diffusion generators. For the special case of chaotic eigenfunctions, this bound can be expressed in terms of certain fourth moments of the vector, yielding a quantitative Fourth Moment Theorem for complex Gaussian approximation on complex Markov diffusion chaos. This extends the results of [ACP14] and [CNPP15] for the real case. Our main ingredients are a complex version of the so called Γ-calculus and Stein's method for the multivariate complex Gaussian distribution.

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Section: Fourth Moment Theorems For Complex Gaussian Approximationmentioning

confidence: 99%