bibkey: “nualart2005multiple” authors: “David Nualart and Giovanni Peccati” year: 2005 title: “Central limit theorems for sequences of multiple stochastic integrals” doi: “10.1214/009117904000000621” url: “https://arxiv.org/pdf/math/0503598v1” claim: “Theorem 1 characterizes Gaussian convergence in a fixed Wiener chaos by vanishing contractions or convergence of the fourth moment, with variance tending to one.” strata_touched: [] license: “citation-only” triage: “anchor”

Gaussian convergence in a fixed Wiener chaos

Theorem 1, page 3 of the cited electronic reprint, fixes a chaos order at least two and assumes variance tending to one. It equates convergence in distribution to a standard Gaussian with convergence of the fourth moment to three, and with vanishing nontrivial contractions.

For a second-chaos spectral sum with square-summable coefficients, the contraction criterion follows from the maximal absolute coefficient tending to zero and a bounded sum of coefficient squares. The countable Gaussian quadratic theorem uses this classical probabilistic mechanism. Its direct characteristic-function proof constructs each actual L² sum, allows row-dependent Gaussian coordinates, and includes zero limiting variance without normalization by that variance. No spectral representation of a prescribed kernel or Wiener integral is asserted by the coefficient theorem. Such an identification is an additional hypothesis of an application of the classical chaos result.

Verified locator

David Nualart and Giovanni Peccati, Annals of Probability 33(1), 177–193 (2005). DOI: 10.1214/009117904000000621. https://arxiv.org/pdf/math/0503598v1, Theorem 1, electronic reprint page 3.