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Preprints, Working Papers, ... Year : 2017

Rates in the Central Limit Theorem and diffusion approximation via Stein's Method

Thomas Bonis

Abstract

We present a way to use Stein's method in order to bound the Wasserstein distance of order $2$ between a measure $\nu$ and another measure $\mu$, assumed to be the reversible measure of a diffusion operator, using non-exchangeable pairs of random variables drawn from $\nu$. We then show that, whenever $\mu$ is the Gaussian measure $\gamma$, one can use exchangeable pairs of random variables drawn from $\nu$ to bound the Wasserstein distance of order $p$, for any $p \geq 2$, between $\nu$ and $\gamma$. Using our results, we are able to obtain convergence rates for the multi-dimensional Central Limit Theorem in terms of Wasserstein distances of order $p \geq 2$. In a second time, we use our approach to bound the Wasserstein distance of order $2$ between the measure of a Markov chain and the reversible measure of a diffusion process satisfying some technical conditions and tackle two problems appearing in the field of data analysis: density estimation for geometric random graphs and sampling via the Langevin Monte Carlo algorithm.
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Dates and versions

hal-01167372 , version 1 (24-06-2015)
hal-01167372 , version 2 (01-07-2016)
hal-01167372 , version 3 (16-05-2017)

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Thomas Bonis. Rates in the Central Limit Theorem and diffusion approximation via Stein's Method. 2017. ⟨hal-01167372v3⟩
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