Wigner chaos and the fourth moment. (English) Zbl 1277.46033

Kemp, Todd; Nourdin, Ivan; Peccati, Giovanni; Speicher, Roland

Ann. Probab. 40, No. 4, 1577-1635 (2012).

The main theorem of this paper is the following central limit theorem (CLT) in the free context, analogous to CLT in the ordinary probabilistic setting (see, e.g., [D. Nualart and G. Peccati, Ann.Probab.33, No. 1, 177–193 (2005; Zbl 1097.60007)]).
{ Theorem A.} Let \(n \geq 2\) and let \(( f_k )_{k \in {\mathbb N}}\) be a sequence of mirror symmetric functions in \(L^2( {\mathbb R}_+^n)\) with \(\| f_k \|_{ L^2( {\mathbb R}_+^n)} = 1\). The following statements are equivalent.

(1)

The fourth moments of the Wigner stochastic integrals \(I_n^S(f_k)\) (with respect to a free Brownian motion \(( S_t )_{ t \geq 0 }\)) converge to 2, i.e., \[ \lim_{k \to \infty} {\mathbb E} [ I_n^S( f_k)^4 ] = 2. \]

(2)

The random variables \(I_n^S(f_k)\) converge in law to the standard semicircular distribution \(S(0,1)\) as \(k \to \infty\).

The proof of Theorem A is through the method of moments which, in the context of the Wigner chaos, is elegantly formulated in terms of noncrossing pairings and partitions. As a matter of fact, a key step toward proving Theorem A is the following characterization result of the fourth moment condition in terms of standard integral contraction operators on the kernels of the stochastic integrals.
{Theorem B.} Let \(n \in {\mathbb N}\), and let \(( f_k )_{k \in {\mathbb N} }\) be a sequence of functions in \(L^2( {\mathbb R}_+^n)\) with \(\| f_k \|_{ L^2( {\mathbb R}_+^n )}\) \(=\) \(1\). The following statements are equivalent.

(1)

The fourth absolute moments of the stochastic integrals \(I_n^S( f_k)\) converge to 2, i.e., \[ \lim_{k \to \infty} {\mathbb E} [ | I_n^S( f_k) |^4 ] = 2. \]

(2)

All nontrivial contractions of \(f_k\) converge to 0: for each \(p=1,2, \dots, n-1\), \[ \lim_{ k \to \infty} f_k \overset {p} {\frown} f_k^* = 0 \quad \text{in} \quad L^2( {\mathbb R}_+^{2n - 2p} ), \] where the \(p\)-th contraction \(f \overset {p} {\frown} g\) of \(f \in L^2( {\mathbb R}_+^n )\) and \(g \in L^2( {\mathbb R}_+^m)\) is the \(L^2(\) \({\mathbb R}_+^{n + m - 2p})\) function defined by nested integration of the middle \(p\) variables in \(f \otimes g\).

By making use of Theorem A together with Theorem B, the authors prove the Wiener-Wigner transfer principle for translating results between the classical and free chaoses.
{Theorem C.} (Wiener-Wigner transfer principle) Let \(n \geq 2\), and let \(( f_k )_{k \in {\mathbb N}}\) be a sequence of fully symmetric functions in \(L^2( {\mathbb R}_+^n)\). Let \(\sigma > 0\) be a finite constant. Then, as \(k \to \infty\):

(1)

\({\mathbb E} [ I_n^W( f_k)^2 ]\) \(\to\) \(n! \sigma^2\) if and only if \({\mathbb E} [ I_n^S ( f_k)^2 ]\) \(\to\) \(\sigma^2\), where \(I_n^W( f)\) denotes the \(n\)-th multiple Wiener-Itô stochastic integral of \(f\) with respect to a standard one-dimensional Brownian motion \(( W_t)_{t > 0}\).

(2)

If the asymptotic relations in (1) are satisfied, then \((I_n^W(f_k))\) converges in law to a normal random variable \(N(0, n! \sigma^2)\) if and only if \((I_n^S(f_k))\) converges in law to a semicircular random variable \(S(0, \sigma^2)\), where the probability distribution \(S(0, \sigma^2)(dx)\) is given by \((2 \pi \sigma^2)^{-1} \sqrt{ 4 \sigma^2 - x^2} dx\) for \(| x | \leqslant 2 \sigma\).

This transfer principle yields immediately a free analog of the Breuer-Major theorem for stationary vectors. Namely,
{ Corollary D.} (Free Breuer-Major theorem) Let \((X_k)_{k \in {\mathbb Z}}\) be a doubly-infinite semicircular system of random variables \(S(0,1)\), and let \(\rho(k)\) \(=\) \(\varphi(X_0 X_k)\) denote the covariance function with \(X_0\). Suppose that there is an integer \(n \geq 1\) such that \[ \sum_{ k \in {\mathbb Z} } | f(k) |^n < \infty. \] Then the sequence \[ V_m = \frac{1}{ \sqrt{m} } \sum_{k=0}^{m-1} U_n ( X_k) \quad \overset {\text{law}} {\longrightarrow} \quad S(0, \sigma^2) \] as \(m \to \infty\), where \(\sigma^2 = \sum_{k \in {\mathbb Z} } \rho(k)^n\), and \(U_n\) is the \(n\)-th Chebyshev polynomial of the second kind.
Another peculiar key point consists in proving some sharp quantitative estimates for the distance to the semicircular law (see below). The key estimate can be obtained by employing Malliavin calculus.
{ Theorem E.} (Key estimate) Let \(S\) be a standard semicircular random variable. Let \(F\) have a finite Wigner chaos expansion, namely, \[ F = \sum_{n=1}^N I_n^S(f_n) \] for some mirror symmetric functions \(f_n \in L^2( {\mathbb R}_+^n)\) and some finite \(N\). When \(h\) has a Fourier expansion as \[ h(x) = \hat{\nu}(x) = \int_{ {\mathbb R} } e^{i x \xi} \nu( d \xi), \] then a seminorm \({\mathcal J}_2(h)\) is defined as \[ {\mathcal J}_2(h) = \int_{ {\mathbb R}} \xi^2 | \nu |( d \xi), \] and \({\mathcal C}_2\) denotes the set of functions \(h\) with \({\mathcal J}_2(h) < \infty\). Then \[ \begin{aligned} d_{ {\mathcal C}_2 } ( F, S) &\equiv \sup _{\substack{ h \in {\mathcal C}_2 \\ {\mathcal J}_2(h) \leqslant 1 }} | {\mathbb E} [ h(F) ] - {\mathbb E} [ h(S) ] | \\ &\leqslant \frac{1}{2} {\mathbb E} \otimes {\mathbb E} \left( \left| \int_0^{\infty} \nabla_t ( N_0^{-1} F) \sharp ( \nabla_t F)^* dt - 1 \otimes 1 \right| \right). \end{aligned} \]
Here \(\nabla\) is the free Cameron-Gross-Malliavin derivative, \(N_0 \equiv \delta \nabla\) is the free Ornstein-Uhlenbeck operator (or free number operator) with the divergence operator \(\delta\), and \(\sharp\) is the product on tensor-product-valued biprocesses in free Malliavin calculus (cf. [P. Biane, Commun.Math.Phys.184, No. 2, 457–474 (1997; Zbl 0874.46049)] and [I. Nourdin and G. Peccati, Normal approximations with Malliavin calculus. From Stein’s method to universality. Cambridge: Cambridge University Press (2012; Zbl 1266.60001)]). For other related works see, e.g., [I. Nourdin and G. Peccati, Interdisciplinary Mathematical Sciences 8, 207–236 (2010; Zbl 1203.60065)]. Especially for Stein’s method, see [I. Nourdin and G. Peccati, Probab.Theory Relat.Fields 145, No. 1–2, 75–118 (2009; Zbl 1175.60053)], and as to some examples of Malliavin calculus applied to central limit theorems, see [D. Nualart and S. Ortiz-Latorre, Stochastic Processes.Appl.118, No. 4, 614–628 (2008; Zbl 1142.60015)].

Reviewer: Isamu Dôku (Saitama)

MSC:

46L54	Free probability and free operator algebras
60H07	Stochastic calculus of variations and the Malliavin calculus
60H30	Applications of stochastic analysis (to PDEs, etc.)

Keywords:

free probability; Wigner chaos; central limit theorem; Malliavin calculus

Citations:

Zbl 1097.60007; Zbl 0874.46049; Zbl 1266.60001; Zbl 1203.60065; Zbl 1175.60053; Zbl 1142.60015

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