### 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.

Original language | English |
---|---|

Pages (from-to) | 1463-1496 |

Number of pages | 34 |

Journal | Bernoulli |

Volume | 24 |

Issue number | 2 |

DOIs | |

Publication status | Published - May 1 2018 |

### Fingerprint

### Keywords

- Convergence in total variation
- Diffusions
- Fourth Moment Theorem
- Invariant measure
- Malliavin calculus
- Multiple stochastic integrals
- Stein's method
- Weak convergence

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Bernoulli*,

*24*(2), 1463-1496. https://doi.org/10.3150/16-BEJ904

**Characterization of the convergence in total variation and extension of the Fourth Moment Theorem to invariant measures of diffusions.** / Kusuoka, Seiichiro; Tudor, Ciprian A.

Research output: Contribution to journal › Article

*Bernoulli*, vol. 24, no. 2, pp. 1463-1496. https://doi.org/10.3150/16-BEJ904

}

TY - JOUR

T1 - Characterization of the convergence in total variation and extension of the Fourth Moment Theorem to invariant measures of diffusions

AU - Kusuoka, Seiichiro

AU - Tudor, Ciprian A.

PY - 2018/5/1

Y1 - 2018/5/1

N2 - 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.

AB - 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.

KW - Convergence in total variation

KW - Diffusions

KW - Fourth Moment Theorem

KW - Invariant measure

KW - Malliavin calculus

KW - Multiple stochastic integrals

KW - Stein's method

KW - Weak convergence

UR - http://www.scopus.com/inward/record.url?scp=85034865678&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85034865678&partnerID=8YFLogxK

U2 - 10.3150/16-BEJ904

DO - 10.3150/16-BEJ904

M3 - Article

AN - SCOPUS:85034865678

VL - 24

SP - 1463

EP - 1496

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 2

ER -