Open Access
November 2012 Chaos of a Markov operator and the fourth moment condition
M. Ledoux
Ann. Probab. 40(6): 2439-2459 (November 2012). DOI: 10.1214/11-AOP685


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.


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M. Ledoux. "Chaos of a Markov operator and the fourth moment condition." Ann. Probab. 40 (6) 2439 - 2459, November 2012.


Published: November 2012
First available in Project Euclid: 26 October 2012

zbMATH: 1266.60042
MathSciNet: MR3050508
Digital Object Identifier: 10.1214/11-AOP685

Primary: 60F05 , 60H99 , 60J35 , 60J60 , 60J99

Keywords: $\Gamma$-calculus , chaos , eigenfunction , fourth moment , iterated gradient , Markov operator , Stein’s method

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 6 • November 2012
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