相关论文: Optimal local central limit theorems on Wiener cha…

We prove a local limit theorem, i.e. a central limit theorem for densities, for a sequence of independent and identically distributed random variables taking values on an abstract Wiener space; the common law of those random variables is…

The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are…

For normalized sums $Z_n$ of i.i.d. random variables, we explore necessary and sufficient conditions which guarantee the normal approximation with respect to the R\'enyi divergence of infinite order. In terms of densities $p_n$ of $Z_n$,…

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…

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…

We establish an unexpected phenomenon of strong regularization along normal convergence on Wiener chaoses. For every sequence of chaotic random variables, convergence in law to the Gaussian distribution is upgraded to superconvergence: 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,…

We consider stochastic wave equations in spatial dimensions $d \geq 4$. We assume that the driving noise is given by a Gaussian noise that is white in time and has some spatial correlation. When the spatial correlation is given by the Riesz…

We investigate the problem of finding necessary and sufficient conditions for convergence in distribution towards a general finite linear combination of independent chi-squared random variables, within the framework of random objects living…

In this work, we study the normal approximation and almost sure central limit theorems for some functionals of an independent sequence of Rademacher random variables. In particular, we provide a new chain rule that improves the one derived…

We establish the convergence of the densities of a sequence of nonlinear functionals of an underlying Gaussian process to the density of a Gamma distribution. The key idea of our work is a new density formula for random variables in the…

We investigate the optimal rate of convergence in the multidimensional normal approximation of vector-valued Wiener-Ito integrals of which components all belong to the same fixed Wiener chaos. Combining Malliavin calculus, Stein's method…

We consider sequences of random variables living in a finite sum of Wiener chaoses. We find necessary and sufficient conditions for convergence in law to a target variable living in the sum of the first two Wiener chaoses. Our conditions…

The Central Limit Theorem states that, in the limit of a large number of terms, an appropriately scaled sum of independent random variables yields another random variable whose probability distribution tends to a stable distribution. The…

We give a new characterization for the convergence in distribution to a standard normal law of a sequence of multiple stochastic integrals of a fixed order with variance one, in terms of the Malliavin derivatives of the sequence. We extend…

We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both…

We combine Malliavin calculus with Stein's method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. We also prove results concerning…

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…

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…

For a compact convex set $K$ and a Poisson point process $\eta$, the union of all Voronoi cells with a nucleus in $K$ is the Poisson-Voronoi approximation of $K$. Lower and upper bounds for the variance and a central limit theorem for the…