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Convergence in $χ^2$ Distance to the Normal Distribution for Sums of Independent Random Variables

Vytas Zacharovas

TL;DR

This work analyzes when the normalized sum $S_n=\frac{X_1+\cdots+X_n}{\sqrt{n}}$ of independent, zero-mean, unit-variance variables converges to the standard normal in the $\chi^2$ distance. The authors develop a Parseval-based framework tied to Hermite polynomials and a Stein-type recurrence to relate $\mathbb{E}H_m(S_n)$ to the moments of the summands, enabling nonasymptotic bounds on $\chi^2(S_n,\mathcal{N})$ in terms of the average $\chi^2(X_j,\mathcal{N})$. They prove that if the average $\frac{1}{n}\sum_j\chi^2(X_j,\mathcal{N})$ is below a constant (0.82 in general, 1.69 under symmetry), then $\chi^2(S_n,\mathcal{N})=O(1/n)$, and symmetry further improves this to $O(1/n^2)$. The paper also connects proximity in $\chi^2$ to subgaussian tail behavior, providing explicit thresholds ensuring a subgaussian condition and thereby strengthening the normal approximation story from both a distance and tail perspective.

Abstract

Suppose $n$ independent random variables $X_1, X_2, \dots, X_n$ have zero mean and equal variance. We prove that if the average of $χ^2$ distances between these variables and the normal distribution is bounded by a sufficiently small constant, then the $χ^2$ distance between their normalized sum and the normal distribution is $O(1/n)$.

Convergence in $χ^2$ Distance to the Normal Distribution for Sums of Independent Random Variables

TL;DR

This work analyzes when the normalized sum of independent, zero-mean, unit-variance variables converges to the standard normal in the distance. The authors develop a Parseval-based framework tied to Hermite polynomials and a Stein-type recurrence to relate to the moments of the summands, enabling nonasymptotic bounds on in terms of the average . They prove that if the average is below a constant (0.82 in general, 1.69 under symmetry), then , and symmetry further improves this to . The paper also connects proximity in to subgaussian tail behavior, providing explicit thresholds ensuring a subgaussian condition and thereby strengthening the normal approximation story from both a distance and tail perspective.

Abstract

Suppose independent random variables have zero mean and equal variance. We prove that if the average of distances between these variables and the normal distribution is bounded by a sufficiently small constant, then the distance between their normalized sum and the normal distribution is .

Paper Structure

This paper contains 7 sections, 17 theorems, 130 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Proofs
  3. $\chi^2$ distance and Hermite polynomials
  4. Recurrent relations for $\chi^2$ distance
  5. Sums of random variables with equal variances
  6. Relation between subgausian condition and $\chi^2$-distance
  7. Appendix: Upper bounds for $C_J(p)$

Key Result

Theorem 1.1

Suppose all the random variables $X_j$ are identically distributed. Then we have $\chi^2(S_n,\mathcal{N})\to 0$ as $n\to\infty$, if and only if $\chi^2(S_n,\mathcal{N})$ is finite for some $n=n_0$, and In this case, $\chi^2$-distance admits and Edgeworth-type expansion which is valid for every $s=3,4,\ldots$ with coefficients $c_j$ representing certain polynomials in the moments $\mathbb{E}X^k$,

Figures (1)

  • Figure 1: Graph of $g(x)$ and $\bigr(g(x)+e^{-2x}g(-x)\bigr)/2$

Theorems & Definitions (32)

  • Theorem 1.1: *bobkov_et_al_2019
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Theorem 2.5
  • ...and 22 more