# Lectures on Gaussian Approximations with Malliavin Calculus

• Ivan Nourdin
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2078)

## Overview.

In a seminal paper of 2005, Nualart and Peccati [40] 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 in distribution to the standard normal law is equivalent to convergence of just the fourth moment. Shortly afterwards, Peccati and Tudor [46] gave a multidimensional version of this characterization.

## Keywords

Brownian Motion Fractional Brownian Motion Hermite Polynomial Fourth Moment Free Probability
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

### Acknowledgements

It is a pleasure to thank the Fondation des Sciences Mathématiques de Paris for its generous support during the academic year 2011–2012 and for giving me the opportunity to speak about my recent research in the prestigious Collège de France. I am grateful to all the participants of these lectures for their assiduity. Also, I would like to warmly thank two anonymous referees for their very careful reading and for their valuable suggestions and remarks. Finally, my last thank goes to Giovanni Peccati, not only for accepting to give a lecture (resulting to the material developed in Sect. 10) but also (and especially!) for all the nice theorems we recently discovered together. I do hope it will continue this way as long as possible!

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