Wiener chaos: moments, cumulants and diagrams

By Giovanni Peccati, Murad S. Taqqu. 2011

282 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/e437c22f7384acbcc5264ea4b61376af1edd1d6f

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In Mathematical Physics

Citations in the corpus

Chaos of a Markov operator and the fourth moment condition (2012, background): “Note, however, that the extension from the finite dimensional Gaussian setting to the infinite dimensional one requires basic analysis on Wiener space as presented, for example, in the first chapter of [18] (see also [22]) in order to fully justify the domain issues and the various conclusions.”
Malliavin-Stein method for variance-gamma approximation on Wiener space (2014, background): “Moreover, a number of combinatorial tools are available to control the moments of such integrals, see [23] for details.”
Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization (2014, unknown): “[26, 37]).”
Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs (2011, background): “Consider the following objects: (i) a collection of Poisson measures η λ , λ > 0 , with σ -ﬁnite and non atomic control measures µ λ such that µ λ ( W ) ∞ and µ λ ( W ) ↑ ∞ , as λ → ∞ , and (ii) a collection of non-diagonal non-negligible symmetric sets H λ ⊂ Z × Z , λ > 0 .”
Moments and Central Limit Theorems for Some Multivariate Poisson Functionals (2012, methodology): “, n ) are not allowed to contain more than one entry from each row, the diagram (π, σ ) is called nonflat in [12].”
Quantitative Breuer-Major Theorems (2010, background): “Proposition 2.1 Let the notation and assumptions of this section prevail (in particular, E [ f ( X 1 )] = 0 and X 1 ∼ N d (0 , I d ) ). α Then, the function f has Hermite rank q ≥ 1 if and only if a α = 0 for all ∈ Λ with | α | < q and a α = 0 for some α ∈ Λ with | α | = q .”
Stochastic analysis for Poisson processes (2014, background): “We again refer the reader to [29] as well as to [22, 15].”
New Berry-Esseen bounds for non-linear functionals of Poisson random measures (2013, background): “Because of the involved structure of the fourth moment of a Poisson multiple integral (resulting from a highly technical so-called diagram formula, see [27]), it is not clear weather a similar result should also be expected for q > 2.”
Multi-Dimensional Gaussian Fluctuations on the Poisson Space (2010, background): “to Peccati and Taqqu [19] or Privault [22] for a complete discussion of multiple Wiener-Itô integrals and their properties (including the forthcoming Proposition 2.”
Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on S^d (2014, unknown): “subordi-nated random ﬁelds In this section, we ﬁrst brieﬂy review some notation and the main results in this area, which we shall deeply exploit in the sequel of the paper.”