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Kac's Central Limit Theorem by Stein's Method

Suprio Bhar, Ritwik Mukherjee, Prathmesh Patil

TL;DR

The paper proves a quantitative central limit theorem for a dependent sequence arising from the angle-doubling map by applying Stein's method to obtain convergence of the normalized sample average $W^f_n$ to $N(0,1)$ in the Wasserstein metric. It first establishes the result for step functions via dependency-neighborhood techniques and then extends to general $f$ with Fourier coefficients decaying as $|a_n|\le M/n^\beta$, $\beta>1/2$, under $\lim_{n\to\infty} \sigma_n^2/n = \sigma^2>0$. This yields a rate $d_W(W^f_n,Z)=O(n^{-1/2})$, providing a stronger form of Kac's original CLT (which was in distribution). The work demonstrates the versatility of Stein's method in ergodic settings and offers a robust tool for CLTs in weakly dependent (Bernoulli-shift-type) processes, with potential applicability to similar dynamical-system–driven sequences.

Abstract

In $1946$, Mark Kac proved a Central Limit type theorem for a sequence of random variables that were not independent. The random variables under consideration were obtained from the angle-doubling map. The idea behind Kac's proof was to show that although the random variables under consideration were not independent, they were what he calls \textit{statistically independent} (in modern terminology, this concept is called long range independence). The final conclusion of his paper was that the sample averages of the random variables, suitably normalized converges to the standard normal distribution. We describe a new proof of Mark Kac's result by applying Stein's method and show that the normalized sample averages converge to the standard normal distribution in the Wasserstein metric, which is stronger than the convergence in distribution.

Kac's Central Limit Theorem by Stein's Method

TL;DR

The paper proves a quantitative central limit theorem for a dependent sequence arising from the angle-doubling map by applying Stein's method to obtain convergence of the normalized sample average to in the Wasserstein metric. It first establishes the result for step functions via dependency-neighborhood techniques and then extends to general with Fourier coefficients decaying as , , under . This yields a rate , providing a stronger form of Kac's original CLT (which was in distribution). The work demonstrates the versatility of Stein's method in ergodic settings and offers a robust tool for CLTs in weakly dependent (Bernoulli-shift-type) processes, with potential applicability to similar dynamical-system–driven sequences.

Abstract

In , Mark Kac proved a Central Limit type theorem for a sequence of random variables that were not independent. The random variables under consideration were obtained from the angle-doubling map. The idea behind Kac's proof was to show that although the random variables under consideration were not independent, they were what he calls \textit{statistically independent} (in modern terminology, this concept is called long range independence). The final conclusion of his paper was that the sample averages of the random variables, suitably normalized converges to the standard normal distribution. We describe a new proof of Mark Kac's result by applying Stein's method and show that the normalized sample averages converge to the standard normal distribution in the Wasserstein metric, which is stronger than the convergence in distribution.
Paper Structure (6 sections, 6 theorems, 35 equations)

This paper contains 6 sections, 6 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Kac's Central Limit Theorem and Main results
  3. A Recall of Preliminary Results and Proof of Theorem \ref{['mt_bmp']}
  4. A Recall of Preliminary Results
  5. Proof of Theorem \ref{['mt_bmp']}
  6. Acknowledgement

Key Result

Theorem 2.1

Let $f:[0,1]\longrightarrow \mathbb{R}$ be an $L^1$ function, with Fourier coefficients satisfying Fourier_condition_f. Let $\{X^f_k\}_{k=0}^{\infty}$ be the collection of random variables as defined by Xf_defn and let $W^f_n$ be as defined by Wnf_defn. Suppose equation sigma_n_lim_assumption is sat

Theorems & Definitions (10)

  • Theorem 2.1
  • Definition 2.2: Rs_St,Villani-book
  • Theorem 2.3: Main result: CLT for Angle Doubling Map
  • Definition 3.1: Rs_St
  • Theorem 3.2: Rs_St
  • Lemma 3.3: Kac_Ber
  • Remark 3.4
  • Lemma 3.5
  • Remark 3.6
  • Lemma 3.7