Abstract
Nourdin and Peccati (Probab Theory Relat Fields 145(1):75–118, 2009) combined the Malliavin calculus and Stein’s method of normal approximation to associate a rate of convergence to the celebrated fourth moment theorem of Nualart and Peccati (Ann Probab 33(1):177–193, 2005). Their analysis, known as the Malliavin-Stein method nowadays, has found many applications towards stochastic geometry, statistical physics and zeros of random polynomials, to name a few. In this article, we further explore the relation between these two fields of mathematics. In particular, we construct exchangeable pairs of Brownian motions and we discover a natural link between Malliavin operators and these exchangeable pairs. By combining our findings with E. Meckes’ infinitesimal version of exchangeable pairs, we can give another proof of the quantitative fourth moment theorem. Finally, we extend our result to the multidimensional case.
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Dedicated to the memory of Charles Stein, in remembrance of his beautiful mind and of his inspiring, creative, very original and deep mathematical ideas, which will, for sure, survive him for a long time.
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Acknowledgements
We would like to warmly thank Christian Döbler and Giovanni Peccati, for very stimulating discussions on exchangeable pairs since the early stage of this work.
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Nourdin, I., Zheng, G. (2019). Exchangeable Pairs on Wiener Chaos. In: Gozlan, N., Latała, R., Lounici, K., Madiman, M. (eds) High Dimensional Probability VIII. Progress in Probability, vol 74. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-26391-1_14
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DOI: https://doi.org/10.1007/978-3-030-26391-1_14
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