Stein's method meets Malliavin calculus: a short survey with new estimates
By Ivan Nourdin, Giovanni Peccati. 2009
65 citations recorded by the source corpus.
We provide an overview of some recent techniques involving the Malliavin calculus of variations and the so-called ``Stein's method'' for the Gaussian approximations of probability distributions. Special attention is devoted to establishing explicit connections with the classic method of moments: in particular, we use interpolation techniques in order to deduce some new estimates for the moments of random variables belonging to a fixed Wiener chaos. As an illustration, a class of central limit theorems associated with the quadratic variation of a fractional Brownian motion is studied in detail.
Subfields: Finance, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/b65d962b13b62e7e660214f49bdbea0c5fa4175d
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Citations in the corpus
- Stein’s density approach and information inequalities (2012, background): “For the origins we refer the reader to [40, 38, 37]; for an overview of the more recent achievements in this field we refer to the monographs [28, 3, 4, 12] or the review articles [27, 31].”
- Stein’s method for invariant measures of diffusions via Malliavin calculus☆ (2012, unknown): “Then using the so-called Stein’s method introduced by Nourdin and Peccati in [7] (see also [9,8]), it is possible to measure the distance between the law of F and the standard normal law N ( 0 , 1 ) .”
- Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants (2011, unknown): “1 (Fourth Moment Theorem – see Nualart and Peccati (2005)). Fix an integer q > 2, and consider a sequence of random variables {Fn : n > 1} belonging to the qth Wiener chaos of X and such that E[F 2 n ] = 1 for all n > 1. Then, as n → ∞, Fn converges in distribution to N ∼ N (0, 1) if and only if E[F”