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Concentration of the empirical measure in Wasserstein distance: bounds involving the covering dimension

Jérôme Dedecker, Aurélie Fischer, Bertrand Michel

Abstract

We give concentration inequalities in Wasserstein distance for the empirical measure of a sequence of independent and identically distributed random variables with values in a Polish space E. These inequalities involve the covering dimension of the support of the distribution of the variables. More precisely, we obtain a complete extension of the concentration inequalities of Fournier and Guillin [2015] in the case where E = R^d , in which the covering dimension replaces the dimension of the ambient space E.

Concentration of the empirical measure in Wasserstein distance: bounds involving the covering dimension

Abstract

We give concentration inequalities in Wasserstein distance for the empirical measure of a sequence of independent and identically distributed random variables with values in a Polish space E. These inequalities involve the covering dimension of the support of the distribution of the variables. More precisely, we obtain a complete extension of the concentration inequalities of Fournier and Guillin [2015] in the case where E = R^d , in which the covering dimension replaces the dimension of the ambient space E.
Paper Structure (17 sections, 15 theorems, 136 equations)

This paper contains 17 sections, 15 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Totally bounded support
  3. Moment inequalities
  4. Hoeffding-type inequalities
  5. Maximal Inequalities and almost sure results
  6. Unbounded support
  7. Fuk-Nagaev type inequalities under weak moment assumptions
  8. Statement of the theorems
  9. Large and moderate deviations under weak moment assumptions
  10. Bernstein-type inequalities under sub/super exponential moments
  11. Proofs in the unbounded cases
  12. Preliminary results
  13. Proof of Theorem \ref{['theo:dev_unbounded-alphageq2p']}
  14. Proof of Theorem \ref{['theo:dev_unbounded-alphaleq2p']}
  15. Proof of Theorem \ref{['theo:dev_unbounded-alphaeq2p']}
  16. ...and 2 more sections

Key Result

Proposition 1

Assume that the support $S$ of $\mu$ is bounded with diameter $\Delta$, and that $N(S, 4^{-(k^*+2)}\Delta)< \infty$ for some integer $k^*\geq 1$. Then, for any $r \geq 2$ and any $p\geq 1$, where $C_p$ only depends on $p$.

Theorems & Definitions (27)

  • Definition 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • Remark 1
  • proof
  • Proposition 4
  • Remark 2
  • ...and 17 more