## Bernoulli

- Bernoulli
- Volume 24, Number 2 (2018), 1463-1496.

### Characterization of the convergence in total variation and extension of the Fourth Moment Theorem to invariant measures of diffusions

Seiichiro Kusuoka and Ciprian A. Tudor

#### Abstract

We give necessary and sufficient conditions to characterize the convergence in distribution of a sequence of arbitrary random variables to a probability distribution which is the invariant measure of a diffusion process. This class of target distributions includes the most known continuous probability distributions. Precisely speaking, we characterize the convergence in total variation to target distributions which are not Gaussian or Gamma distributed, in terms of the Malliavin calculus and of the coefficients of the associated diffusion process. We also prove that, among the distributions whose associated squared diffusion coefficient is a polynomial of second degree (with some restrictions on its coefficients), the only possible limits of sequences of multiple integrals are the Gaussian and the Gamma laws.

#### Article information

**Source**

Bernoulli, Volume 24, Number 2 (2018), 1463-1496.

**Dates**

Received: November 2015

Revised: May 2016

First available in Project Euclid: 21 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1505980901

**Digital Object Identifier**

doi:10.3150/16-BEJ904

**Mathematical Reviews number (MathSciNet)**

MR3706799

**Zentralblatt MATH identifier**

06778370

**Keywords**

convergence in total variation diffusions Fourth Moment Theorem invariant measure Malliavin calculus multiple stochastic integrals Stein’s method weak convergence

#### Citation

Kusuoka, Seiichiro; Tudor, Ciprian A. Characterization of the convergence in total variation and extension of the Fourth Moment Theorem to invariant measures of diffusions. Bernoulli 24 (2018), no. 2, 1463--1496. doi:10.3150/16-BEJ904. https://projecteuclid.org/euclid.bj/1505980901