UNIVERSITY
OF WROC£AW
 
Main Page
Contents
Online First
General Information
Instructions for authors


VOLUMES
43.2 43.1 42.2 42.1 41.2 41.1 40.2
40.1 39.2 39.1 38.2 38.1 37.2 37.1
36.2 36.1 35.2 35.1 34.2 34.1 33.2
33.1 32.2 32.1 31.2 31.1 30.2 30.1
29.2 29.1 28.2 28.1 27.2 27.1 26.2
26.1 25.2 25.1 24.2 24.1 23.2 23.1
22.2 22.1 21.2 21.1 20.2 20.1 19.2
19.1 18.2 18.1 17.2 17.1 16.2 16.1
15 14.2 14.1 13.2 13.1 12.2 12.1
11.2 11.1 10.2 10.1 9.2 9.1 8
7.2 7.1 6.2 6.1 5.2 5.1 4.2
4.1 3.2 3.1 2.2 2.1 1.2 1.1
 
 
WROC£AW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

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.

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

P.D. Miller, Applied Asymptotic Analysis, Grad. Stud. Math. 75, Amer. Math. Soc., Providence, RI, 2006.

I. Nourdin and G. Peccati, Stein’s method on Wiener chaos, Probab. Theory Related Fields 145 (2009), 75-118.

I. Nourdin and G. Peccati, Normal Approximations with Malliavin Calculus, Cambridge Tracts in Math. 192, Cambridge Univ. Press, Cambridge, 2012.

D. Nualart, The Malliavin Calculus and Related Topics, 2nd ed., Probab. Appl. (New York), Springer, Berlin, 2006.

D. Nualart and S. Ortiz-Latorre, Central limit theorems for multiple stochastic integrals and Malliavin calculus, Stochastic Process. Appl. 118 (2008), 614-628.

D. Nualart and G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005), 177-193.

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.

Download:    Abstract    Full text