Invariance Principles for Homogeneous Sums: Universality of Gaussian Wiener Chaos

By Ivan Nourdin, Giovanni Peccati, Gesine Reinert. 2009

119 citations recorded by the source corpus.

We compute explicit bounds in the normal and chi-square approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. Our techniques combine an invariance principle by Mossel, O'Donnell and Oleszkiewicz with a refinement of some recent results by Nourdin and Peccati, about the approximation of laws of random variables belonging to a fixed (Gaussian) Wiener chaos. In particular, we show that chaotic random variables enjoy the following form of \textsl{universality}: (a) the normal and chi-square approximations of any homogenous sum can be completely characterized and assessed by first switching to its Wiener chaos counterpart, and (b) the simple upper bounds and convergence criteria available on the Wiener chaos extend almost verbatim to the class of homogeneous sums. These results partially rely on the notion of ``low influences'' for functions defined on product spaces, and provide a generalization of central and non-central limit theorems proved by Nourdin, Nualart and Peccati. They also imply a further drastic simplification of the method of moments and cumulants -- as applied to the proof of probabilistic limit theorems -- and yield substantial generalizations, new proofs and new insights into some classic findings by de Jong and Rotar'. Our tools involve the use of Malliavin calculus, and of both the Stein's method and the Lindeberg invariance principle for probabilistic approximations.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/03528ace8f4166809a35ebe10eb78d0cd95c4d3d

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Citations in the corpus

Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants (2011, unknown): “2 (Fourth Moment Bounds – see [9, 15]) Fix q > 2, let F be an element of the qth Wiener chaos of X with unit variance, and let N ∼ N (0, 1).”
Multi-Dimensional Gaussian Fluctuations on the Poisson Space (2010, methodology): “The bounds involving functionals of a Gaussian process come from [9], whereas those for Poisson functionals are taken from Theorem 4.”
Stein's method meets Malliavin calculus: a short survey with new estimates (2009, methodology): “2 (See [22]) Fix d > 2, and let C = (Cij)16i,j6d be a d×d covariance matrix.”
Quantitative Breuer-Major Theorems (2010, background): “The next statement, which is the main result of the paper, asserts that the quantities deﬁned above can be used to write explicit bounds of the type (1.5).”
Malliavin-Stein method for variance-gamma approximation on Wiener space (2014, methodology): “The universality phenomenon for homogeneous sums has been addressed by Rotar [26] and later also by Nourdin, Peccati and Reinert [16], who consider especially multivariate extensions in case of normal and Gamma limiting distributions by means of Stein’s method and Malliavin calculus.”
Cumulants on the Wiener Space (2009, unknown): “multivariate case, in relation with quasi-sure analysis; [16] deals with infinite-dimensional second order Poincaré inequalities; in [19], one can find new explicit expressions for the density of functionals of Gaussian field as well as new concentration inequalities (see also [3] for some applicati”
The optimal fourth moment theorem (2013, background): “1 has sparkled a great amount of generalisations and applications, ranging from density estimates [18] to entropic CLTs [17], and from estimates for Gaussian polymers [21] to universality results [16].”
New Berry-Esseen bounds for non-linear functionals of Poisson random measures (2013, background): “…4] (most prominently the fractional Brownian motion [15]), spherical random fields [1, 14], random matrix theory [17] and universality questions [19], whereas the findings for Poisson random measures have attracted applications in stochastic geometry [10, 13, 36, 37], non-parametric Bayesian…”