A. Inspired by the insightful article [4], we revisit the Nualart-Peccati-criterion [13] (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion generators. We are not only able to drastically simplify all of its previous proofs, but also to provide new settings of diffusive generators (Laguerre, Jacobi) where such a criterion holds. Convergence towards gamma and beta distributions under moment conditions is also discussed.

Let Y = (Y (t)) t≥0 be a zero-mean Gaussian stationary process with covariance function ρ : R → R satisfying ρ(0) = 1. Let f : R → R be a square-integrable function with respect to the standard Gaussian measure, and suppose the Hermite rank of f is d ≥ 1. If R |ρ(s)| d ds < ∞, then the celebrated Breuer-Major theorem (in its continuous version) asserts that the finite-dimensional distributions of Z ε := √ ε ·/ε 0 f (Y (s))ds converge to those of σW as ε → 0, where W is a standard Brownian motion and σ is some explicit constant. Since its first appearance in 1983, this theorem has become a crucial probabilistic tool in different areas, for instance in signal processing or in statistical inference for fractional Gaussian processes.The goal of this paper is twofold. Firstly, we investigate the tightness in the Breuer-Major theorem. Surprisingly, this problem did not receive a lot of attention until now, and the best available condition due to Ben Hariz [1] is neither arguably very natural, nor easy-to-check in practice. In contrast, our condition very simple, as it only requires that |f | p must be integrable with respect to the standard Gaussian measure for some p strictly bigger than 2. It is obtained by means of the Malliavin calculus, in particular Meyer inequalities.Secondly, and motivated by a problem of geometrical nature, we extend the continuous Breuer-Major theorem to the notoriously difficult case of self-similar Gaussian processes which are not necessarily stationary. An application to the fluctuations associated with the length process of a regularized version of the bifractional Browninan motion concludes the paper.where H q (x) is the qth Hermite polynomial.It has become a central result in modern stochastic analysis that, under the conditionconverge, as ε tends to zero, to those of σW , where W = (W (t)) t≥0 is a standard Brownian

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We prove a version of the multidimensional Fourth Moment Theorem for chaotic random vectors, in the general context of diffusion Markov generators. In addition to the usual componentwise convergence and unlike the infinite-dimensional Ornstein-Uhlenbeck generator case, another moment-type condition is required to imply joint convergence of of a given sequence of vectors.2000 Mathematics Subject Classification. 60F05, 60J35, 60J99.

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.

Abstract. We prove a quantitative Fourth Moment Theorem for Wigner integrals of any order with symmetric kernels, generalizing an earlier result from Kemp et al. (2012). The proof relies on free stochastic analysis and uses a new biproduct formula for bi-integrals. A consequence of our main result is a Nualart-Ortiz-Latorre type characterization of convergence in law to the semicircular distribution for Wigner integrals. As an application, we provide Berry-Esseen type bounds in the context of the free Breuer-Major theorem for the free fractional Brownian motion.

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