Simple Bounds for the Convergence of Empirical and Occupation Measures in 1-Wasserstein Distance

By Emmanuel Boissard. 2011

80 citations recorded by the source corpus.

We study the problem of non-asymptotic deviations between a reference measure and its empirical version, in the 1-Wasserstein metric, under the standing assumption that the reference measure satisfies a transport-entropy inequality. We extend some results of F. Bolley, A. Guillin and C. Villani with simple proofs. Our methods are based on concentration inequalities and extend to the general setting of measures on a Polish space. Deviation bounds for the occupation measure of a contracting Markov chain in 1-Wasserstein distance are also given. Throughout the text, several examples are worked out, including the cases of Gaussian measures on separable Banach spaces, and laws of diffusion processes.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/64c4173bb94ae360384c63360fa04170e07bd037

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In Applied Mathematics

In Statistics and Probability

Citations in the corpus

On the rate of convergence of empirical measures in $\infty$-transportation distance (2014, background): “[4] E. Boissard, Simple bounds for convergence of empirical and occupation measures in 1-Wasserstein distance, Electron.”
On the rate of convergence in Wasserstein distance of the empirical measure (2013, unknown): “Unfortunately, the construction of Boissard-Le Gouic, based on iterative trees, was a little too complicated to yield sharp rates.”
On the mean speed of convergence of empirical and occupation measures in Wasserstein distance (2011, background): “There are many works in statistics devoted to convergence rates in some metric associated with the weak convergence of measures, see e.g. the book of A. Van der Vaart and J. Wellner [30].”
Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices (2012, result): “These results have been extended in [Boi11] and [BLG11].”
Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance (2017, result): “btaining tail bounds for the quantity Wp(µ,µˆn). The arguments of [BGV07] rely on transportation inequalities such as the celebrated Bobkov-G¨otze inequality [BG99]. These arguments were simpliﬁed in [Boi11], but, as noted in [BLG14], the analysis becomes much easier if the development of tail bound”
The derivation of swarming models: Mean-field limit and Wasserstein distances (2013, background): “…≤ C , for all t ∈ [0 , T ], where C is independent of N . the k k We next substitute ρ N ( t ) for ρ ( t ) in proof of Theorem 3 ) follows from [15, Proposition 1.2 of Annexe A] (see also [14, 17]) that P (cid:0) X N, 0 : d 1 ( µ 0 N , ρ 0 ) ≥ ε ≤ CN − s ′ , where C and s ′ are positive constants.”
Convergence and concentration of empirical measures under Wasserstein distance in unbounded functional spaces (2018, background): “In the case of p = 1, it provides a better concentration rate compared to [6] and [16] under the sub-exponential tail condition.”