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Lectures on Gaussian Approximations with Malliavin Calculus

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Séminaire de Probabilités XLV

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,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.

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Notes

  1. 1.

    You may watch the videos of the lectures at http://www.sciencesmaths-paris.fr/index.php?page=175.

  2. 2.

    Any adapted process u that is either càdlàg or càglàd admits a progressively measurable version. We will always assume that we are dealing with it.

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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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  1. Institut de Mathématiques Élie Cartan, Université de Lorraine, 70239, 54506, Vandoeuvre-lès-Nancy Cedex, France

    Ivan Nourdin

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  1. , Laboratoire de Mathématiques, Université de Versailles-St Quentin, avenue des États-Unis 45, Versailles, 78035, France

    Catherine Donati-Martin

  2. IECN, Campus scientifique, BP 239, Nancy-Université, INRIA, Vandoeuvre-lès-Nancy, 54506, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St-Quentin, avenue des Etats-Unis 45, Versailles, 78035, France

    Alain Rouault

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Nourdin, I. (2013). Lectures on Gaussian Approximations with Malliavin Calculus. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLV. Lecture Notes in Mathematics(), vol 2078. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00321-4_1

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