Nualart-Peccati准则的另一个证明
Yet another proof of the Nualart-Peccati criterion
摘要：
在2005年，Nualart和Peccati令人惊讶地证明了多个维纳-伊托积分的归一化序列在分布上的依分布收敛与标准高斯律的收敛等价于四阶矩收敛于3。最近，这一结果已被推广到一个序列的多重维格纳积分，在自由布朗运动的背景下。本文的目的是提供一个基本的，统一的证明这两个结果。唯一先进的，必要的工具是多重积分的乘积公式。除了这个公式，其余的证明只依赖于软组合参数。
英文摘要：
In 2005, Nualart and Peccati showed that, surprisingly, the convergence in distribution of a normalized sequence of multiple Wiener-Itô integrals towards a standard Gaussian law is equivalent to convergence of just the fourth moment to 3. Recently, this result has been extended to a sequence of multiple Wigner integrals, in the context of free Brownian motion. The goal of the present paper is to offer an elementary, unifying proof of these two results. The only advanced, needed tool is the product formula for multiple integrals. Apart from this formula, the rest of the proof only relies on soft combinatorial arguments.
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