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Contents of PMS, Vol. 42, Fasc. 2,
pages 177 - 194
DOI: 10.37190/0208-4147.00060
Published online 7.10.2022
 

Generalizations of the fourth moment theorem

N. Naganuma

Abstract:

Azmoodeh {et al.} established a criterion regarding convergence of the and even moments of random variables in a Wiener chaos with fixed order guaranteeing the central convergence of the random variables. This was a major step in studies of the fourth moment theorem. In this paper, we provide further generalizations of the fourth moment theorem by building on their ideas. More precisely, further criteria implying central convergence are provided: (i) the convergence of the and even moment, (ii) the convergence of the and even moments.

2010 AMS Mathematics Subject Classification: Primary 60F05; Secondary 33C45, 60H07.

Keywords and phrases: the fourth moment theorem, Nualart--Peccati criterion, central convergence, Wiener chaos.

References


E. Azmoodeh, S. Campese, and G. Poly, Fourth moment theorems for Markov diffusion generators, J. Funct. Anal. 266 (2014), 2341-2359.

E. Azmoodeh, D. Malicet, G. Mijoule, and G. Poly, Generalization of the Nualart–Peccati criterion, Ann. Probab. 44 (2016), 924-954.

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F.W.J. Olver, D.W. Lozier, R.F. Boisvert, and C.W. Clark (eds.), NIST Handbook of Mathematical Functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC, and Cambridge Univ. Press, Cambridge, 2010.

G. Peccati and C.A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals, in: Séminaire de Probabilités XXXVIII, Lecture Notes in Math. 1857, Springer, Berlin, 2005, 247–262.

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