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

• Berry–Esseen theorem for non-i.i.d. variables is an extension of the classical result that quantifies the error in normal approximation with explicit bounds for heterogeneous and dependent structures.
• It employs methodologies such as Fourier analysis, Stein's method, coupling, and dependency graphs to derive optimal rates under various moment conditions and distance metrics.
• The framework enhances error estimates for complex models including U-statistics, local dependence, and weakly dependent time series, aiding precise probabilistic approximations.

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 (Vershynin, 5 Feb 2026, Pinelis, 2013, Klartag et al., 2010).

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 (Vershynin, 5 Feb 2026).

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 $\mathbb{E} X_j = 0$0 (Pinelis, 2013).

• Total Variation and Entropic Bounds: For independent but non-identically distributed random variables with bounded entropic distance $\mathbb{E} X_j = 0$1, Bobkov–Chistyakov–Götze (Bobkov et al., 2011) showed

$\mathbb{E} X_j = 0$2

with stronger norm but same rate in the Lyapunov ratio.

• $\mathbb{E} X_j = 0$3 and Rényi Distances: If the $\mathbb{E} X_j = 0$4 have densities with matching moments to the Gaussian up to order $\mathbb{E} X_j = 0$5, in the polynomial-density case,

$\mathbb{E} X_j = 0$6

thereby gaining a rate improvement by moment matching (Delplancke et al., 2017).

• Chaos Expansions: General functionals and U-statistics of independent variables, including non-i.i.d. cases, yield

$\mathbb{E} X_j = 0$7

recovering classical rates under standard moment hypotheses (Privault et al., 2020).

3. Dependency Graphs and General Dependent Structures

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

• Dependency Graphs: Consider $\mathbb{E} X_j = 0$8 with means $\mathbb{E} X_j = 0$9, variances $\mathrm{Var}(X_j) = \sigma_j^2 > 0$0, and finite $\mathrm{Var}(X_j) = \sigma_j^2 > 0$1-th moments ($\mathrm{Var}(X_j) = \sigma_j^2 > 0$2), and a dependency graph $\mathrm{Var}(X_j) = \sigma_j^2 > 0$3 of maximal degree $\mathrm{Var}(X_j) = \sigma_j^2 > 0$4 such that disjoint non-neighbor sets index independent subfamilies. Define

$\mathrm{Var}(X_j) = \sigma_j^2 > 0$5

with $\mathrm{Var}(X_j) = \sigma_j^2 > 0$6.

The Kolmogorov distance satisfies (Janisch et al., 2022)

$\mathrm{Var}(X_j) = \sigma_j^2 > 0$7

with explicit constants for all $\mathrm{Var}(X_j) = \sigma_j^2 > 0$8, improving over Stein-method bounds with much worse degree dependence.

The classical rate $\mathrm{Var}(X_j) = \sigma_j^2 > 0$9 is immediately recovered when $\mathbb{E}|X_j|^3 < \infty$0.

• Local Dependence: Under a local dependency structure (e.g., dependency neighborhoods/blocks), the error decomposes with explicit control via local indices, e.g.,

$\mathbb{E}|X_j|^3 < \infty$1

where $\mathbb{E}|X_j|^3 < \infty$2 and $\mathbb{E}|X_j|^3 < \infty$3 encode the local graph-neighborhood sizes (Cai et al., 2 Feb 2026).

4. Weak Dependence, Markov Chains, and Dynamical Systems

• Stationary Sequences and Time Series: For strictly stationary mean-zero processes, under a weak-dependence coefficient $\mathbb{E}|X_j|^3 < \infty$4 satisfying $\mathbb{E}|X_j|^3 < \infty$5,

$\mathbb{E}|X_j|^3 < \infty$6

for $\mathbb{E}|X_j|^3 < \infty$7, thus interpolating between independence ($\mathbb{E}|X_j|^3 < \infty$8, $\mathbb{E}|X_j|^3 < \infty$9) and more weakly dependent cases (Jirak, 2016).

• Spectral Methods: For weakly dependent Markov/non-Markov sequences with controlled cumulants and spectral gap, non-uniform Berry–Esseen bounds and Edgeworth expansions yield

$$\sup_{x \in \mathbb{R}} \left| \mathbb{P}\left( \frac{S_n}{(\sum \sigma_j^2)^{1/2}} \le x \right) - \Phi(x) \right| \le C \frac{\sum_j \mathbb{E}|X_j|^3}{(\sum_j \sigma_j^2)^{3/2}}$$0

and analogues for transport metrics and higher-order approximations (Hafouta, 2022).

• Polynomial Densities and Markovian Recursion: For independent, non-identically distributed $$\sup_{x \in \mathbb{R}} \left| \mathbb{P}\left( \frac{S_n}{(\sum \sigma_j^2)^{1/2}} \le x \right) - \Phi(x) \right| \le C \frac{\sum_j \mathbb{E}|X_j|^3}{(\sum_j \sigma_j^2)^{3/2}}$$1 with density admitting a Hermite expansion and matching moments up to order $$\sup_{x \in \mathbb{R}} \left| \mathbb{P}\left( \frac{S_n}{(\sum \sigma_j^2)^{1/2}} \le x \right) - \Phi(x) \right| \le C \frac{\sum_j \mathbb{E}|X_j|^3}{(\sum_j \sigma_j^2)^{3/2}}$$2, and subject to polynomial-density growth conditions, Markovian spectral analysis gives (Delplancke et al., 2017)

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

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 (Goldstein, 2010) and exchangeable pairs (even with unbounded jumps) (Shao et al., 2017), allow explicit Kolmogorov and Wasserstein bounds for wide classes including dependency graphs.

For exchangeable pairs $$\sup_{x \in \mathbb{R}} \left| \mathbb{P}\left( \frac{S_n}{(\sum \sigma_j^2)^{1/2}} \le x \right) - \Phi(x) \right| \le C \frac{\sum_j \mathbb{E}|X_j|^3}{(\sum_j \sigma_j^2)^{3/2}}$$4 with regression condition $$\sup_{x \in \mathbb{R}} \left| \mathbb{P}\left( \frac{S_n}{(\sum \sigma_j^2)^{1/2}} \le x \right) - \Phi(x) \right| \le C \frac{\sum_j \mathbb{E}|X_j|^3}{(\sum_j \sigma_j^2)^{3/2}}$$5,

$$\sup_{x \in \mathbb{R}} \left| \mathbb{P}\left( \frac{S_n}{(\sum \sigma_j^2)^{1/2}} \le x \right) - \Phi(x) \right| \le C \frac{\sum_j \mathbb{E}|X_j|^3}{(\sum_j \sigma_j^2)^{3/2}}$$6

controls the rate, with all terms explicit (Shao et al., 2017). 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

$$\sup_{x \in \mathbb{R}} \left| \mathbb{P}\left( \frac{S_n}{(\sum \sigma_j^2)^{1/2}} \le x \right) - \Phi(x) \right| \le C \frac{\sum_j \mathbb{E}|X_j|^3}{(\sum_j \sigma_j^2)^{3/2}}$$7

with $$\sup_{x \in \mathbb{R}} \left| \mathbb{P}\left( \frac{S_n}{(\sum \sigma_j^2)^{1/2}} \le x \right) - \Phi(x) \right| \le C \frac{\sum_j \mathbb{E}|X_j|^3}{(\sum_j \sigma_j^2)^{3/2}}$$8 the first projection in the Hoeffding decomposition (Privault et al., 2020, Cai et al., 2 Feb 2026).

• Weighted Sums: For sums with non-uniform weights,

$$\sup_{x \in \mathbb{R}} \left| \mathbb{P}\left( \frac{S_n}{(\sum \sigma_j^2)^{1/2}} \le x \right) - \Phi(x) \right| \le C \frac{\sum_j \mathbb{E}|X_j|^3}{(\sum_j \sigma_j^2)^{3/2}}$$9

thus providing sharp error control for linear combinations (Privault et al., 2020, Klartag et al., 2010).

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

$\Phi$0

with $\Phi$1 the Lyapunov ratio. Pinelis (Pinelis, 2013) and refinements (Pinelis, 2011) showed exponential decay in tails by combining Stein’s equation with Chen–Shao concentration,

$\Phi$2

where $\Phi$3 and $\Phi$4 are parameters, allowing optimization for small or large deviations.

• Multivariate Extensions: For independent, non-identically distributed vectors in $\Phi$5, if $\Phi$6, the error for convex sets $\Phi$7 satisfies

$\Phi$8

with explicit constants (Raič, 2018).

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References (arXiv IDs)

• (Janisch et al., 2022) Berry-Esseen-type estimates for random variables with a sparse dependency graph
• (Cai et al., 2 Feb 2026) Refined Berry-Esseen bounds under local dependence
• (Privault et al., 2020) Berry-Esseen bounds for functionals of independent random variables
• (Jirak, 2016) Berry-Esseen theorems under weak dependence
• (Pinelis, 2013) On the nonuniform Berry--Esseen bound
• (Bobkov et al., 2011) Berry-Esseen bounds in the entropic central limit theorem
• (Pinelis, 2011) Improved nonuniform Berry--Esseen-type bounds
• (Vershynin, 5 Feb 2026) A friendly proof of the Berry-Esseen theorem
• (Klartag et al., 2010) Variations on the Berry-Esseen theorem
• (Goldstein, 2010) A Berry-Esseen bound with applications to vertex degree counts in the Erdős-Rényi random graph
• (Raič, 2018) A multivariate Berry--Esseen theorem with explicit constants
• (Hafouta, 2022) Non-uniform Berry-Esseen theorem and Edgeworth expansions with applications to transport distances for weakly dependent random variables
• (Delplancke et al., 2017) Berry-Esseen bounds for the chi-square distance in the Central Limit Theorem: a Markovian approach
• (Shao et al., 2017) Berry-Esseen Bounds of Normal and Non-normal Approximation for Unbounded Exchangeable Pairs
• (Dey et al., 2022) Berry-Esseen Theorem for Sample Quantiles with Locally Dependent Data

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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.