相关论文: Fisher Information and the Fourth Moment Theorem

In this paper, we first establish general bounds on the Fisher information distance to the class of normal distributions of Malliavin differentiable random variables. We then study the rate of Fisher information convergence in the central…

An Edgeworth-type expansion is established for the relative Fisher information distance to the class of normal distributions of sums of i.i.d. random variables, satisfying moment conditions. The validity of the central limit theorem is…

We compute the exact rates of convergence in total variation associated with the 'fourth moment theorem' by Nualart and Peccati (2005), stating that a sequence of random variables living in a fixed Wiener chaos verifies a central limit…

A moderate deviation principle as well as moderate and large deviation inequalities for a sequence of elements living inside a fixed Wiener chaos associated with an isonormal Gaussian process are shown. The conditions under which the…

We prove that a normalized sequence of multiple Wigner integrals (in a fixed order of free Wigner chaos) converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2,…

Nualart & Pecatti ([Nualart and Peccati, 2005, Thm 1]) established the first fourth-moment theorem for random variables in a fixed Wiener chaos, i.e. they showed that convergence of the sequence of fourth moments to the fourth moment of the…

In this work, we establish conditions ensuring convergence in distribution of a sequence admitting a Wiener-It\^o chaos representation to a nondegenerate Gaussian measure on a separable Hilbert space. Our first main result shows that,…

In a seminal paper of 2005, Nualart and Peccati discovered a surprising central limit theorem (called the "Fourth Moment Theorem" in the sequel) for sequences of multiple stochastic integrals of a fixed order: in this context, convergence…

A new information-theoretic approach to the central limit theorem for stable laws is presented. The main novelty is the concept of relative fractional Fisher information, which shares most of the properties of the classical one, included…

We develop a general method to study the Fisher information distance in central limit theorem for nonlinear statistics. We first construct completely new representations for the score function. We then use these representations to derive…

In this paper, we consider the distribution-dependent SDE driven by fractional Brownian motion with small noise and study the rate of Fisher information convergence in the central limit theorem for the solution of SDE, then we show that the…

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 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…

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…

The fourth moment theorem provides error bounds of the order $\sqrt{{\mathbb E}(F^4) - 3}$ in the central limit theorem for elements $F$ of Wiener chaos of any order such that ${\mathbb E}(F^2) = 1$. It was proved by Nourdin and Peccati…

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 2005, Nualart and Peccati showed that, surprisingly, the convergence in distribution of a normalized sequence of multiple Wiener-It\^o integrals towards a standard Gaussian law is equivalent to convergence of just the fourth moment to 3.…

We give conditions for an O(1/n) rate of convergence of Fisher information and relative entropy in the Central Limit Theorem. We use the theory of projections in L2 spaces and Poincare inequalities, to provide a better understanding of the…

The celebrated Nualart-Peccati criterion [Ann. Probab. 33 (2005) 177-193] ensures the convergence in distribution toward a standard Gaussian random variable $N$ of a given sequence $\{X_n\}_{n\ge1}$ of multiple Wiener-It\^{o} integrals of…

We show that the mathematical form of the information measure of Fisher's I for a Gibbs' canonical probability distribution (the most important one in statistical mechanics) incorporates important features of the intrinsic structure of…