Related papers: Wigner chaos and the fourth moment
We review and present some known results for non-linear functionals of Gaussian variables in the context of discrete Gaussian fields defined on the $d$ dimensional lattice. Our main result is a Central Limit Theorem in the spirit of the…
In 2005, Nualart and Peccati showed the so-called Fourth Moment Theorem asserting that, for a sequence of normalized multiple Wiener-It\^o integrals to converge to the standard Gaussian law, it is necessary and sufficient that its fourth…
In \cite{BNT}, a framework to prove almost sure central limit theorems for sequences $(G_n)$ belonging to the Wiener space was developed, with a particular emphasis of the case where $G_n$ takes the form of a multiple Wiener-It\^o integral…
Consider $F$ an element of the second Wiener chaos with variance one. In full generality, we show that, for every integer $p\ge 1$, there exists $\eta_p>0$ such that if $\kappa_4(F)<\eta_p$ then the Malliavin derivative of $F$ admits a…
We revisit the central limit theorem for integrated periodograms, equivalently for Toeplitz quadratic forms of stationary Gaussian sequences. Under a regular-variation assumption allowing long-memory singularities and slowly varying…
We analyze from the viewpoint of an abstract Markov operator recent results by Nualart and Peccati, and Nourdin and Peccati, on the fourth moment as a condition on a Wiener chaos to have a distribution close to Gaussian. In particular, we…
In two new papers (Bierme et al., 2013) and (Nourdin and Peccati, 2015), sharp general quantitative bounds \ are given to complement the well-known fourth moment theorem of Nualart and Peccati, by which a sequence in a fixed Wiener chaos…
In the framework of classical probability, we consider the normal product distribution $F_\infty \sim N_1 \times N_2$ where $N_1, N_2$ are two independent standard normal random variable, and in the setting of free probability, $F_\infty…
A quantitative "fourth moment theorem" is provided for any self-adjoint element in a homogeneous Wigner chaos: the Wasserstein distance is controlled by the distance from the fourth moment to two. The proof uses the free counterpart of the…
We revisit the moment method to obtain a slightly strengthened version of the usual semicircular law. Our version assumes only that the upper triangular entries of Hermitian random matrices are independent, have mean zero and variances…
In this paper, we prove four-moment theorems for multidimensional free Poisson limits on free Wigner chaos or the free Poisson algebra. We prove that, under mild technical conditions, a bi-indexed sequence of free stochastic integrals in…
In the large-$N$, classical limit, the Bose-Hubbard dimer undergoes a transition to chaos when its tunnelling rate is modulated in time. We use exact and approximate numerical simulations to determine the features of the dynamically…
We prove that an adequately rescaled sequence $\{F_n\}$ of self-adjoint operators, living inside a fixed free Wigner chaos of even order, converges in distribution to a centered free Poisson random variable with rate $\lambda>0$ if and only…
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…
We prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. Our techniques rely on a…
We prove an exact fourth moment bound for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result -- that has been elusive for several years -- shows that the so-called…
We have studied statistical properties of the values of the Wigner function W(x) of 1D quantum maps on compact 2D phase space of finite area V. For this purpose we have defined a Wigner function probability distribution P(w) = (1/V) int…
In [NP09a], Nourdin and Peccati established a neat characterization of Gamma approximation on a fixed Wiener chaos in terms of convergence of only the third and fourth cumulants. In this paper, we investigate the rate of convergence in…
We study rates of convergence in central limit theorems for the partial sum of squares of general Gaussian sequences, using tools from analysis on Wiener space. No assumption of stationarity, asymptotically or otherwise, is made. The main…
Let {F_n} be a normalized sequence of random variables in some fixed Wiener chaos associated with a general Gaussian field, and assume that E[F_n^4] --> E[N^4]=3, where N is a standard Gaussian random variable. Our main result is the…