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Generalization of the Nualart-Peccati criterion. (English) Zbl 1342.60026
Let \(\{X_n\}_{n\geq 1}\) be a sequence of multiple Wiener-Itô integrals of a fixed order \(p\geq 2\) such that \(\mathbb E[X_n^2]\rightarrow 1\) as \(n\rightarrow\infty\). The fourth moment theorem, due to D. Nualart and G. Peccati [Ann. Probab. 33, No. 1, 177–193 (2005; Zbl 1097.60007)] asserts that \(X_n\) converges in law to a standard normal distribution if and only if \(\mathbb E[X_n^4]\rightarrow 3\) as \(n\rightarrow\infty\). This result offers a drastic simplification of the method of moments in this setting.
The main result of the present paper is a generalisation of this fourth moment theorem. The authors prove that, for \(\{X_n\}_{n\geq 1}\) as above and a fixed integer \(k\geq 2\), \(X_n\) converges in law to a standard normal distribution if and only if \(\mathbb E[X_n^{2k}]\rightarrow\mathbb E[N^{2k}]\), where \(N\sim\mathrm N(0,1)\).
The authors go further and give an explicit rate of convergence in the total variation distance in this setting, in the spirit of [I. Nourdin and G. Peccati, Probab. Theory Relat. Fields 145, No. 1-2, 75–118 (2009; Zbl 1175.60053)]: \[ d_{TV}(X_n,\mathrm N(0,1))\leq C_k\sqrt{\frac{\mathbb E[X_n^{2k}]}{\mathbb E[N^{2k}]}-1}, \] where an explicit expression for the constant \(C_k\) is given.
Several new inequalities for multiple Wiener-Itô integrals arise from the proofs, and may be of some independent interest. In fact, the above results on multiple Wiener-Itô integrals follow from more general and powerful results on eigenfunctions of a diffusive Markov operator. Finally, the paper concludes with two conjectures relevant to this framework, which would make interesting areas for future research.

MSC:
60F05 Central limit and other weak theorems
60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
47D07 Markov semigroups and applications to diffusion processes
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
34L05 General spectral theory of ordinary differential operators
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References:
[1] Abramowitz, M. and Stegun, I. (1972). Handbook of Mathematical Functions : With Formulas , Graphs , and Mathematical Tables . Courier Dover Publications, New York. · Zbl 0543.33001
[2] Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions. Encyclopedia of Mathematics and Its Applications 71 . Cambridge Univ. Press, Cambridge.
[3] Azmoodeh, E., Campese, S. and Poly, G. (2014). Fourth moment theorems for Markov diffusion generators. J. Funct. Anal. 266 2341-2359. · Zbl 1292.60078
[4] Azmoodeh, E., Peccati, G. and Poly, G. (2014). Convergence towards linear combinations of chi-squared random variables: A Malliavin-based approach. Available at . arXiv:1409.5551 · Zbl 1337.60018
[5] Bakry, D. (1994). L’hypercontractivité et son utilisation en théorie des semigroupes. In Lectures on Probability Theory ( Saint-Flour , 1992). Lecture Notes in Math. 1581 1-114. Springer, Berlin. · Zbl 0856.47026
[6] Bakry, D., Gentil, I. and Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 348 . Springer, Cham. · Zbl 1376.60002
[7] Bouleau, N. and Hirsch, F. (1991). Dirichlet Forms and Analysis on Wiener Space. De Gruyter Studies in Mathematics 14 . de Gruyter, Berlin. · Zbl 0748.60046
[8] Breuer, P. and Major, P. (1983). Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 425-441. · Zbl 0518.60023
[9] Carbery, A. and Wright, J. (2001). Distributional and \(L^{q}\) norm inequalities for polynomials over convex bodies in \(\mathbb{R}^{n}\). Math. Res. Lett. 8 233-248. · Zbl 0989.26010
[10] Chambers, D. and Slud, E. (1989). Central limit theorems for nonlinear functionals of stationary Gaussian processes. Probab. Theory Related Fields 80 323-346. · Zbl 04518605
[11] Chen, L. H. (2014). Stein meets Malliavin in normal approximation. Available at . arXiv:1407.5172 · Zbl 1322.60028
[12] Chen L., H. and Poly, G. (2015). Stein’s method, Malliavin calculus, Dirichlet forms and the fourth moment theorem. In Festschrift Masatoshi Fukushima (Z.-Q. Chen, N. Jacob, M. Takeda andT. Uemura, eds.). Interdisciplinary Mathematical Sciences 17 107-130. World Scientific, Singapore. · Zbl 1343.60012
[13] Granville, A. and Wigman, I. (2011). The distribution of the zeros of random trigonometric polynomials. Amer. J. Math. 133 295-357. · Zbl 1218.60042
[14] Kemp, T., Nourdin, I., Peccati, G. and Speicher, R. (2012). Wigner chaos and the fourth moment. Ann. Probab. 40 1577-1635. · Zbl 1277.46033
[15] Ledoux, M. (2012). Chaos of a Markov operator and the fourth moment condition. Ann. Probab. 40 2439-2459. · Zbl 1266.60042
[16] Nazarov, F. and Sodin, M. (2011). Fluctuations in random complex zeroes: Asymptotic normality revisited. Int. Math. Res. Not. IMRN 24 5720-5759. · Zbl 1242.60051
[17] Nourdin, I. and Peccati, G. (2009). Noncentral convergence of multiple integrals. Ann. Probab. 37 1412-1426. · Zbl 1171.60323
[18] Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos. Probab. Theory Related Fields 145 75-118. · Zbl 1175.60053
[19] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus : From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192 . Cambridge Univ. Press, Cambridge. · Zbl 1266.60001
[20] Nourdin, I. and Peccati, G. (2014). The optimal fourth moment theorem. Proc. Amer. Math. Soc. · Zbl 1317.60021
[21] Nourdin, I., Peccati, G., Poly, G. and Simone, R. (2014). Classical and free fourth moment theorems: Universality and thresholds. Available at . arXiv:1407.6216 · Zbl 1356.60037
[22] Nourdin, I., Peccati, G. and Swan, Y. (2014). Entropy and the fourth moment phenomenon. J. Funct. Anal. 266 3170-3207. · Zbl 1292.94010
[23] Nourdin, I. and Poly, G. (2013). Convergence in total variation on Wiener chaos. Stochastic Process. Appl. 123 651-674. · Zbl 1259.60029
[24] Nualart, D. (2006). The Malliavin Calculus and Related Topics , 2nd ed. Probability and Its Applications ( New York ). Springer, Berlin. · Zbl 1099.60003
[25] Nualart, D. and Ortiz-Latorre, S. (2008). Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 614-628. · Zbl 1142.60015
[26] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 177-193. · Zbl 1097.60007
[27] Peccati, G. (2014). Quantitative CLTs on a Gaussian space: A survey of recent developments. In Journées MAS 2012. ESAIM Proc. 44 61-78. EDP Sci., Les Ulis. · Zbl 1327.60062
[28] Peccati, G. and Tudor, C. A. (2005). Gaussian limits for vector-valued multiple stochastic integrals. In Séminaire de Probabilités XXXVIII. Lecture Notes in Math. 1857 247-262. Springer, Berlin. · Zbl 1063.60027
[29] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 293 . Springer, Berlin. · Zbl 0917.60006
[30] Serre, D. (2010). Matrices : Theory and Applications , 2nd ed. Graduate Texts in Mathematics 216 . Springer, New York. · Zbl 1008.15002
[31] Surgailis, D. (2003). CLTs for polynomials of linear sequences: Diagram formula with illustrations. In Theory and Applications of Long-Range Dependence 111-127. Birkhäuser, Boston, MA. · Zbl 1032.60017
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