Chaos of a Markov operator and the fourth moment condition
By Michel Ledoux. 2012
60 citations recorded by the source corpus.
We analyze from the viewpoint of an abstract Markov operator recent results by Nualart and Peccati, and Nourdin and Peccati, on the fourth moment as a condition on a Wiener chaos to have a distribution close to Gaussian. In particular, we are led to introduce a notion of chaos associated to a Markov operator through its iterated gradients and present conditions on the (pure) point spectrum for a sequence of chaos eigenfunctions to converge to a Gaussian distribution. Convergence to gamma distributions may be examined similarly.
Subfields: Finance, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/6794168c83aabaa5ede2003dc409357a5c904a00
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Citations in the corpus
- Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on S^d (2014, unknown): “Such results are currently being generalized in several directions, including Poisson processes, free probability, random matrices, Markov subordinators and information theory (see [3], [12], [24], [27], [28] e.”
- Fourth Moment Theorems for Markov Diffusion Generators (2013, methodology): “The deVnition of "general chaos" given in [4] makes use of iterated gradients and, although including the Ornstein-Uhlenbeck case, prevents one to reach the Laguerre and Jacobi structures.”
- Convergence in total variation on Wiener chaos (2012, unknown): “(i) extension to the multidimensional case; (ii) improvement of the rate of convergence; (iii) extension to other types of chaoses (in the spirit of [9]).”