The Annals of Probability

Four moments theorems on Markov chaos

Solesne Bourguin, Simon Campese, Nikolai Leonenko, and Murad S. Taqqu

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We obtain quantitative four moments theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is that it admits four moments. These results are obtained by first proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions, which is of independent interest, and making use of the new concept of chaos grade. For the heavy-tailed Pearson distributions, this seems to be the first time that sufficient conditions in terms of (finitely many) moments are given in order to converge to a distribution that is not characterized by its moments.

Article information

Ann. Probab., Volume 47, Number 3 (2019), 1417-1446.

Received: December 2017
Revised: March 2018
First available in Project Euclid: 2 May 2019

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J99: None of the above, but in this section

Markov operator diffusion generator Gamma calculus Pearson distributions Stein’s method limit theorems


Bourguin, Solesne; Campese, Simon; Leonenko, Nikolai; Taqqu, Murad S. Four moments theorems on Markov chaos. Ann. Probab. 47 (2019), no. 3, 1417--1446. doi:10.1214/18-AOP1287.

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