---
title: Berry–Esseen Theorem for Non-i.i.d. Variables
url: https://www.emergentmind.com/topics/berry-esseen-theorem-for-non-i-i-d-random-variables
type: topic
---

# Berry–Esseen Theorem for Non-i.i.d. Variables

The Berry-Esseen theorem provides explicit quantitative rates for the central limit theorem by bounding the distance between the distribution function of a standardized sum of random variables and the standard normal distribution. The extension of this theorem to non-i.i.d. random variables is a mature area, covering a diverse range of dependence structures, moment conditions, and metrics, including Kolmogorov, total variation, relative entropy, and chi-square distances. Principle methodologies include Fourier analysis, Stein's method, coupling and concentration, dependency graphs, and chaos/expansion approaches.

## 1. Classical Berry–Esseen Bound and the Non-i.i.d. Setting

For a sum \( S_n = X_1 + \cdots + X_n \) of independent, not necessarily identically distributed random variables with \( \mathbb{E} X_j = 0 \), \( \mathrm{Var}(X_j) = \sigma_j^2 > 0 \), and \( \mathbb{E}|X_j|^3 < \infty \), the Berry–Esseen theorem asserts

\[
\sup_{x \in \mathbb{R}} \left| \mathbb{P}\left( \frac{S_n}{(\sum \sigma_j^2)^{1/2}} \le x \right) - \Phi(x) \right| \le C \frac{\sum_j \mathbb{E}|X_j|^3}{(\sum_j \sigma_j^2)^{3/2}}
\]

where \( \Phi \) is the standard normal cdf and \( C \) is an absolute constant, with sharp estimates \( 0.4748 < C < 0.5600 \) known for the constant in the non-i.i.d. case [2602.06234][1301.2828][1002.3970].

This form is robust: the dependence on the Lyapunov ratio is known to be optimal in order. The proof via Fourier analysis (Esseen's smoothing inequality) extends directly to the non-i.i.d. case, exploiting independence for factorization of characteristic functions [2602.06234].

## 2. Refinements: Non-i.i.d. Sums and Optimal Constants

Classical extensions of the Berry–Esseen theorem for non-i.i.d. sequences include results in Kolmogorov, total variation, entropy, and \( \chi^2 \) metrics:

- **Kolmogorov Distance (Uniform and Non-uniform):** The nonuniform Berry–Esseen bound strengthens the error in central and tail zones as
  \[
  |F_n(x) - \Phi(x)| \le C_{nu} \frac{\sum_i \mathbb{E}|X_i|^3}{(\sum_i \sigma_i^2)^{3/2}} \frac{1}{1 + x^3}
  \]
  with currently best-proven general-case bound \( C_{nu} \le 31.9 \) [1301.2828].

- **Total Variation and Entropic Bounds:** For independent but non-identically distributed random variables with bounded entropic distance \( D(X_j) \), Bobkov–Chistyakov–Götze [1105.4119] showed
  \[
  \| F_n-\Phi \|_{TV} \le C(D) \sum_j \mathbb{E}|X_j|^3 \quad \text{and} \quad D(S_n \| N(0,1)) \le C'(D) \sum_j \mathbb{E}|X_j|^4
  \]
  with stronger norm but same rate in the Lyapunov ratio.

- **\(\chi^2\) and Rényi Distances:** If the \( X_j \) have densities with matching moments to the Gaussian up to order \( r \), in the polynomial-density case,
  \[
  \chi^2(\mathcal{L}(S_n), N(0,1)) \le C n^{-(r-1)/2}
  \]
  thereby gaining a rate improvement by moment matching [1709.09410].

- **Chaos Expansions:** General functionals and U-statistics of independent variables, including non-i.i.d. cases, yield
  \[
  d_K\left( \frac{S_n}{\sigma_n}, N \right) \le C \left( \sum_j \mathbb{E}|X_j|^4 \right)^{1/2}/\sigma_n^2
  \]
  recovering classical rates under standard moment hypotheses [2010.04387].

## 3. Dependency Graphs and General Dependent Structures

A significant direction concerns summands with weak dependencies encoded via a sparsity graph.

- **Dependency Graphs:** Consider \( \{X_i\} \) with means \( \mu_i \), variances \( \sigma_i^2 \), and finite \( \delta \)-th moments (\( \delta > 2 \)), and a dependency graph \( G=(V,E) \) of maximal degree \( \Delta \) such that disjoint non-neighbor sets index independent subfamilies. Define
  \[
  \xi_\delta = (n/A_\delta)^{1/\delta} \sqrt{ \sigma^2 / (n(\Delta+1)) }
  \]
  with \( A_\delta = \sum_i \mathbb{E}|X_i - c_i|^\delta \).

  The Kolmogorov distance satisfies [2212.02590]
  \[
  \sup_z \left| \mathbb{P}\left( \frac{S_n}{\sigma} \le z \right) - \Phi(z) \right| \le \max \left\{ C_1(\delta) \xi_\delta^{-\delta/(\delta+1)} \left( \frac{\Delta+1}{n} \right)^{(\delta-2)/(2(\delta+1))}, C_2 \xi_3^{-3}\left( \frac{\Delta+1}{n} \right)^{1/2} \right\}
  \]
  with explicit constants for all \( \delta > 2 \), improving over Stein-method bounds with much worse degree dependence.

  The classical rate \( O(n^{-1/2}) \) is immediately recovered when \( \Delta=0 \).

- **Local Dependence:** Under a local dependency structure (e.g., dependency neighborhoods/blocks), the error decomposes with explicit control via local indices, e.g.,
  \[
  \sup_z |P(W \le z) - \Phi(z)| \le C \left\{ \frac{\kappa^2}{\sigma^3} \sum_{i=1}^n \|X_i\|_4^3 + \frac{\kappa^{1/2} (\kappa + \tau^{1/2})}{\sigma^2 (\sum \|X_i\|_4^4)^{1/2}} \right\}
  \]
  where \( \kappa \) and \( \tau \) encode the local graph-neighborhood sizes [2602.02217].

## 4. Weak Dependence, Markov Chains, and Dynamical Systems

- **Stationary Sequences and Time Series:** For strictly stationary mean-zero processes, under a weak-dependence coefficient \( \delta_p(l) \) satisfying \( \sum l^2 \delta_p(l) < \infty \),
  \[
  \sup_{x \in \mathbb{R}} \left| P \left( \frac{S_n}{\sqrt{n s_n^2}} \le x \right) - \Phi(x) \right| \le C n^{p/2-1}
  \]
  for \( 2 < p \le 3 \), thus interpolating between independence (\( p=3 \), \( n^{-1/2} \)) and more weakly dependent cases [1606.01617].

- **Spectral Methods:** For weakly dependent Markov/non-Markov sequences with controlled cumulants and spectral gap, non-uniform Berry–Esseen bounds and Edgeworth expansions yield
  \[
  \left| P(W_n \le x) - \Phi(x) \right| \le C (1+|x|)^{-m} \sigma_n^{-1}
  \]
  and analogues for transport metrics and higher-order approximations [2210.07204].

- **Polynomial Densities and Markovian Recursion:** For independent, non-identically distributed \( X_i \) with density admitting a Hermite expansion and matching moments up to order \( r \), and subject to polynomial-density growth conditions, Markovian spectral analysis gives [1709.09410]
  \[
  \chi^2(\mathcal{L}(S_n), N(0,1)) \le C n^{-(r-1)/2}
  \]
  which is optimal under these hypotheses.

## 5. Stein’s Method, Exchangeable Pairs, and Coupling

- **Stein's Method for Local and Global Dependence:** Both size-bias coupling [1005.4390] and exchangeable pairs (even with unbounded jumps) [1710.01554], allow explicit Kolmogorov and Wasserstein bounds for wide classes including dependency graphs.

  For exchangeable pairs \( (W, W') \) with regression condition \( \mathbb{E}[W'-W|W]=-\lambda(W+R) \),
  \[
  \sup_{z} |P(W \le z) - \Phi(z)| \le \mathbb{E}\left| 1 - \frac{1}{2\lambda} \mathbb{E}[(W'-W)^2|W] \right| + \frac{1}{2\lambda}\mathbb{E}[|W'-W|^2] + \mathbb{E}|R|
  \]
  controls the rate, with all terms explicit [1710.01554]. This approach is robust to dependent structures provided an appropriate pair and remainder can be constructed.

## 6. Non-i.i.d. Rates for Functionals and U-Statistics

- **Hoeffding Decomposable U-Statistics:** Berry–Esseen bounds extend to degenerate and non-degenerate U-statistics of independent non-identically distributed random variables, given by
  \[
  d_K\Bigl( \frac{U_n-\theta}{\sqrt{\operatorname{Var}(U_n)}}, N \Bigr) \le C \frac{\sum_i \mathbb{E}|V_i|^3}{ (\operatorname{Var}(U_n))^{3/2} }
  \]
  with \( V_i \) the first projection in the Hoeffding decomposition [2010.04387][2602.02217].

- **Weighted Sums:** For sums with non-uniform weights,
  \[
  d_K\Bigl( \frac{S_n}{ (\sum w_i^2 \sigma_i^2)^{1/2} }, N \Bigr) \le C \frac{ \sum |w_i|^3 \mathbb{E}|X_i|^3 }{ (\sum w_i^2 \sigma_i^2)^{3/2} }
  \]
  thus providing sharp error control for linear combinations [2010.04387][1002.3970].

## 7. Non-uniform and High-dimensional Berry–Esseen Bounds

- **Non-uniform Bounds in Tails:** For the non-i.i.d. case, best-known results provide
  \[
  | F_n(x) - \Phi(x) | \le C_{nu} L_n (1 + x^3)^{-1}
  \]
  with \( L_n \) the Lyapunov ratio. Pinelis [1301.2828] and refinements [1109.0741] showed exponential decay in tails by combining Stein’s equation with Chen–Shao concentration,
  \[
  |\mathsf{P}(S > z) - \mathsf{P}(Z > z)| \le A \frac{\sum g(\xi_i/v)}{e^{\theta z}}
  \]
  where \( g(x) = x^2 \wedge |x|^3 \) and \( v, \theta \) are parameters, allowing optimization for small or large deviations.

- **Multivariate Extensions:** For independent, non-identically distributed vectors in \( \mathbb{R}^d \), if \( \sum \mathrm{Var}(X_i) = I_d \), the error for convex sets \( A \) satisfies
  \[
  | P(W \in A) - N(0, I_d)\{A\} | \le (42 d^{1/4} + 16) \sum_i \mathbb{E}|X_i|^3
  \]
  with explicit constants [1802.06475].

---

### References (arXiv IDs)
- [2212.02590] Berry-Esseen-type estimates for random variables with a sparse dependency graph
- [2602.02217] Refined Berry-Esseen bounds under local dependence
- [2010.04387] Berry-Esseen bounds for functionals of independent random variables
- [1606.01617] Berry-Esseen theorems under weak dependence
- [1301.2828] On the nonuniform Berry--Esseen bound
- [1105.4119] Berry-Esseen bounds in the entropic central limit theorem
- [1109.0741] Improved nonuniform Berry--Esseen-type bounds
- [2602.06234] A friendly proof of the Berry-Esseen theorem
- [1002.3970] Variations on the Berry-Esseen theorem
- [1005.4390] A Berry-Esseen bound with applications to vertex degree counts in the Erdős-Rényi random graph
- [1802.06475] A multivariate Berry--Esseen theorem with explicit constants
- [2210.07204] Non-uniform Berry-Esseen theorem and Edgeworth expansions with applications to transport distances for weakly dependent random variables
- [1709.09410] Berry-Esseen bounds for the chi-square distance in the Central Limit Theorem: a Markovian approach
- [1710.01554] Berry-Esseen Bounds of Normal and Non-normal Approximation for Unbounded Exchangeable Pairs
- [2208.09072] Berry-Esseen Theorem for Sample Quantiles with Locally Dependent Data

---

The modern theory of the Berry–Esseen theorem for non-i.i.d. random variables encompasses a hierarchy of models from independence with heterogeneous distributions, local/global dependences (dependency graphs, block structures), weak/mixing dependencies (Markov/dynamical systems), and non-linear functionals (U-statistics, vertex counts, quantiles). Across these settings, sharp rates are controlled by moment conditions, explicit dependency/coupling indices, and, in many cases, obtain optimal constants or tail-adaptive rates for both uniform and tail (non-uniform) distances to normality.

Source: https://www.emergentmind.com/topics/berry-esseen-theorem-for-non-i-i-d-random-variables