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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.

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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