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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: 28 (0 UL)Continuous Breuer-Major theorem: tightness and non-stationarity Campese, Simon ; Nourdin, Ivan ; in Annals of Probability (2020), 48(1), 147-177 Detailed reference viewed: 207 (9 UL)Four Moments Theorems on Markov Chaos ; Campese, Simon ; et al in 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: 91 (2 UL)Multivariate Gaussian approxi- mations on Markov chaoses Campese, Simon ; Nourdin, Ivan ; Peccati, Giovanni et al in Electronic Communications in Probability (2016), 21 Detailed reference viewed: 229 (8 UL)Approximate Normality of High-Energy Hyperspherical Eigenfunctions Campese, Simon ; ; Rossi, Maurizia E-print/Working paper (2015) Detailed reference viewed: 121 (8 UL)Optimal rates, Fourth Moment Theorems and non-linear functionals of Brownian local times Campese, Simon Doctoral thesis (2014) The present dissertation provides contributions to three distinct topics of modern stochastic analysis, namely: (a) optimal rates of convergence in multidimensional central limit theorems (CLTs) on a ... [more ▼] The present dissertation provides contributions to three distinct topics of modern stochastic analysis, namely: (a) optimal rates of convergence in multidimensional central limit theorems (CLTs) on a Gaussian space, (b) Fourth Moment Theorems (and associated multidimensional generalisations) in the framework of the chaos of a Markov generator and (c) CLTs for non-linear functionals of Brownian local times. [less ▲] Detailed reference viewed: 330 (42 UL)Fourth Moment Theorems for Markov diffusion generators Azmoodeh, Ehsan ; Campese, Simon ; Poly, Guillaume Joseph in Journal of Functional Analysis (2014), 266(4), 23412359 Inspired by the insightful article [4], we revisit the Nualart–Peccati criterion [13] (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion ... [more ▼] Inspired by the insightful article [4], we revisit the Nualart–Peccati criterion [13] (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion generators. We are not only able to drastically simplify all of its previous proofs, but also to provide new settings of diffusive generators (Laguerre, Jacobi) where such a criterion holds. Convergence towards Gamma and Beta distributions under moment conditions is also discussed. [less ▲] Detailed reference viewed: 181 (27 UL)Optimal Convergence Rates and One-Term Edgeworth Expansions for Multidimensional Functionals of Gaussian Fields Campese, Simon in ALEA: Latin American Journal of Probability and Mathematical Statistics (2013) We develop techniques for determining the exact asymptotic speed of convergence in the multidimensional normal approximation of smooth functions of Gaussian fields. As a by-product, our findings yield ... [more ▼] We develop techniques for determining the exact asymptotic speed of convergence in the multidimensional normal approximation of smooth functions of Gaussian fields. As a by-product, our findings yield exact limits and often give rise to one-term generalized Edgeworth expansions increasing the speed of convergence. Our main mathematical tools are Malliavin calculus, Stein's method and the Fourth Moment Theorem. This work can be seen as an extension of the results of arXiv:0803.0458 to the multi-dimensional case, with the notable difference that in our framework covariances are allowed to fluctuate. We apply our findings to exploding functionals of Brownian sheets, vectors of Toeplitz quadratic functionals and the Breuer-Major Theorem. [less ▲] Detailed reference viewed: 69 (4 UL) |
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