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A Proof of the Central Limit Theorem Using the $2$-Wasserstein Metric

Calvin Wooyoung Chin

Abstract

We prove the Lindeberg--Feller central limit theorem without using characteristic functions or Taylor expansions, but instead by measuring how far a distribution is from the standard normal distribution according to the $2$-Wasserstein metric. This falls under the category of renormalization group methods. The facts we need about the metric are explained and proved in detail. We illustrate the idea on a classical version of the central limit theorem before going into the main proof.

A Proof of the Central Limit Theorem Using the $2$-Wasserstein Metric

Abstract

We prove the Lindeberg--Feller central limit theorem without using characteristic functions or Taylor expansions, but instead by measuring how far a distribution is from the standard normal distribution according to the -Wasserstein metric. This falls under the category of renormalization group methods. The facts we need about the metric are explained and proved in detail. We illustrate the idea on a classical version of the central limit theorem before going into the main proof.
Paper Structure (6 sections, 17 theorems, 55 equations)

This paper contains 6 sections, 17 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Proof of a version of the Lindeberg--Lévy CLT
  3. Inverses of CDFs and the $2$-Wasserstein metric
  4. Proof of Theorem \ref{['thm:average_closer']}
  5. Proof of the Lindeberg--Feller CLT
  6. Measure theoretic probabilistic proofs

Key Result

Proposition 0

Let $X,X_1,X_2,\dots$ be random variables with mean $0$ and variance $1$. If $W_2(X_n,X) \Rightarrow 0$, then $X_n \Rightarrow X$.

Theorems & Definitions (26)

  • Proposition 0
  • Proposition 0
  • Theorem 1
  • Theorem 2: Lindeberg--Feller
  • Theorem 3: Lindeberg--Lévy, bounded, lacunary
  • Lemma 3
  • Lemma 3
  • proof : Proof of Theorem \ref{['thm:lindeberg_levy_bounded_lacunary']}
  • Proposition 3
  • Corollary 3
  • ...and 16 more