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Normal approximation for partial sums: general convex costs

Jérôme Dedecker, Florence Merlevède, Emmanuel Rio

Abstract

We provide non-asymptotic bounds and asymptotic limits for convex transport costs between the distribution of partial sums of independent and identically distributed square integrable and centered random variables and the normal distribution with mean zero and the same variance. The proof relies on controlling the transport cost by an appropriate ideal distance, combined with an adaptation of Lindeberg's method. The numerical constants and the asymptotic constants are explicit.

Normal approximation for partial sums: general convex costs

Abstract

We provide non-asymptotic bounds and asymptotic limits for convex transport costs between the distribution of partial sums of independent and identically distributed square integrable and centered random variables and the normal distribution with mean zero and the same variance. The proof relies on controlling the transport cost by an appropriate ideal distance, combined with an adaptation of Lindeberg's method. The numerical constants and the asymptotic constants are explicit.
Paper Structure (19 sections, 18 theorems, 245 equations)

This paper contains 19 sections, 18 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Normal approximation for convex costs
  4. Coupling inequalities
  5. Proofs
  6. Preliminary considerations
  7. Proof of Theorem \ref{['Thmkappa']}
  8. Proof of Remark \ref{['remark2']}
  9. Proof of Theorem \ref{['coravecphi']}
  10. Proof of Remark \ref{['remTh22']}
  11. Proof of Proposition \ref{['coravecxp']}
  12. Proof of Corollaries \ref{['asymptoticconstants']} and \ref{['asymptotickappa1g']}
  13. Proof of Theorem \ref{['UpperBoundHLtail']}
  14. Proof of Corollary \ref{['corweakmoments']}
  15. Proof of Corollary \ref{['asymptoticconstantsforLambda']}
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let ${\mathcal{F}}_x:= \{ f : {\mathbb R} \rightarrow {\mathbb R} \, : \, \Vert f' \Vert_{\infty} \leq x \, , \, f' \text{ is $1$-Lipschitz} \}$. Then

Theorems & Definitions (28)

  • Proposition 2.1
  • Theorem 2.1
  • Remark 2.1
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.2
  • Remark 2.3
  • Proposition 2.2
  • Corollary 2.1
  • Remark 2.4
  • ...and 18 more