Properties of convergence in Dirichlet structures

Malicet, Dominique; Poly, Guillaume

doi:10.1016/j.jfa.2013.02.007

Cited by 6 publications

(7 citation statements)

References 6 publications

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“…Secondly, let us quote the paper [11], in which necessary and sufficient conditions are given (in term of the absolute continuity of the laws) so that the classical Central Limit Theorem holds in total variation. Finally, let us mention [1] or [6] for conditions ensuring the convergence in total variation for random variables in Sobolev or Dirichlet spaces. Although all the above examples are related to very different frameworks, they have in common the use of a particular structure of the involved variables; loosely speaking, this structure allows to derive a kind of "non-degeneracy" in an appropriate sense which, in turn, enables to reinforce the convergence, from the Fortet-Mourier distance to the total variation one.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Convergence in Law Implies Convergence in Total Variation for Polynomials in Independent Gaussian, Gamma or Beta Random Variables

Nourdin

Poly

2016

Progress in Probability

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Consider a sequence of polynomials of bounded degree evaluated in independent Gaussian, Gamma or Beta random variables. We show that, if this sequence converges in law to a nonconstant distribution, then (i) the limit distribution is necessarily absolutely continuous with respect to the Lebesgue measure and (ii) the convergence automatically takes place in the total variation topology. Our proof, which relies on the Carbery-Wright inequality and makes use of a diffusive Markov operator approach, extends the results of [9] to the Gamma and Beta cases.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Convergence in Law Implies Convergence in Total Variation for Polynomials in Independent Gaussian, Gamma or Beta Random Variables

Nourdin

Poly

2016

Progress in Probability

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“…Moreover we prove that if a sequence F n , n ∈ N, is bounded in every D 3,p , p ≥ 1, lim n F n = F in L 2 and det σ F > 0 a.s., then lim n F n = F in total variation. Recently, Malicet and Poly [13] have proved an alternative version of this result: if lim n F n = F in D 1,2 and det σ F > 0 a.s. then the convergence takes place in the total variation distance. The paper is organized as follows.…”

Section: Introductionmentioning

confidence: 97%

On the distances between probability density functions

Bally

Caramellino

2014

Electron. J. Probab.

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We give estimates of the distance between the densities of the laws of two functionals F and G on the Wiener space in terms of the Malliavin-Sobolev norm of F − G. We actually consider a more general framework which allows one to treat with similar (Malliavin type) methods functionals of a Poisson point measure (solutions of jump type stochastic equations). We use the above estimates in order to obtain a criterion which ensures that convergence in distribution implies convergence in total variation distance; in particular, if the functionals at hand are absolutely continuous, this implies convergence in L 1 of the densities.We define now the functional spaces and the differential operators.Simple functionals. A random variable F is called a simple functional if there exists f ∈ C ∞ p (R J ) such that F = f (V ). We denote through S the set of simple functionals. Simple processes. A simple process is a random variable U = (U 1 , . . . , U J ) in R J such that U i ∈ S for each i ∈ {1, . . . , J}. We denote by P the space of the simple processes. On P we define the scalar product ·, · : P × P → S, (U, V ) → U,

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“…We first recall that κ 4,m (c) = κ 4 (Φ m (c, Z)) with Z k , k ∈ N, standard normal. So, the identity (B.3) is proved in [23] for iterated integrals and remains true for stochastic series because 4) is straightforward (but see also formula (13) in [22]) and (B.5) appears in [22] and [19]. (B.6) is an immediate consequence of (B.4)-(B.5) and (B.3).…”

Section: Lemma B1mentioning

confidence: 85%

An invariance principle for stochastic series I. Gaussian limits

Bally,

Caramellino

2015

Preprint

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We study invariance principles and convergence to a Gaussian limit for stochastic series of the form S(c, Z) = ∞ m=1 α1<...<αm c(α 1 , ..., α m ) m i=1 Z αi where Z k , k ∈ N is a sequence of centred independent random variables of unit variance. In the case when the Z k 's are Gaussian, S(c, Z) is an element of the Wiener chaos and convergence to a Gaussian limit (so the corresponding nonlinear CLT) has been intensively studied by Nualart, Peccati, Nourdin and several other authors. The invariance principle consists in taking Z k with a general law. It has also been considered in the literature, starting from the seminal papers of Jong, and a variety of applications including U -statistics are of interest. Our main contribution is to study the convergence in total variation distance and to give estimates of the error.

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Properties of convergence in Dirichlet structures

Cited by 6 publications

References 6 publications

Convergence in Law Implies Convergence in Total Variation for Polynomials in Independent Gaussian, Gamma or Beta Random Variables

Convergence in Law Implies Convergence in Total Variation for Polynomials in Independent Gaussian, Gamma or Beta Random Variables

On the distances between probability density functions

An invariance principle for stochastic series I. Gaussian limits

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