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   Approximation of Hilbert-valued Gaussians on Dirichlet structures; Campese, Simon  in Electronic Journal of Probability (2020), 25 We introduce a framework to derive quantitative central limit theorems in the context of non-linear approximation of Gaussian random variables taking values in a separable Hilbert space. In particular ... [more ▼] We introduce a framework to derive quantitative central limit theorems in the context of non-linear approximation of Gaussian random variables taking values in a separable Hilbert space. In particular, our method provides an alternative to the usual (non-quantitative) finite dimensional distribution convergence and tightness argument for proving functional convergence of stochastic processes. We also derive four moments bounds for Hilbert-valued random variables with possibly infinite chaos expansion, which include, as special cases, all finite-dimensional four moments results for Gaussian approximation in a diffusive context proved earlier by various authors. Our main ingredient is a combination of an infinite-dimensional version of Stein’s method as developed by Shih and the so-called Gamma calculus. As an application, rates of convergence for the functional Breuer-Major theorem are established. [less ▲] Detailed reference viewed: 30 (0 UL)Four Moments Theorems on Markov Chaos; Campese, Simon ; et alin Annals of Probability (2019), 47(3), 1417-1446 We obtain quantitative four moments theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is ... [more ▼] We obtain quantitative four moments theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is that it admits four moments. These results are obtained by first proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions, which is of independent interest, and making use of the new concept of chaos grade. For the heavy-tailed Pearson distributions, this seems to be the first time that sufficient conditions in terms of (finitely many) moments are given in order to converge to a distribution that is not characterized by its moments. [less ▲] Detailed reference viewed: 100 (2 UL) Gaussian approximations of nonlinear statistis on the sphere; ; et al in Journal of Mathematical Analysis and Applications (2016), 436(2), 1121-1148 Detailed reference viewed: 141 (8 UL)Semicircular limits on the free Poisson chaos: counterexamples to a transfer principle; Peccati, Giovanni in Journal of Functional Analysis (2014), 267(4), Detailed reference viewed: 123 (0 UL) |
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