Berry–Esseen Bounds: Rates and Extensions

Updated 1 June 2026

Berry–Esseen bounds are quantitative measures of convergence rates in central limit theorems, defined using moment conditions and structural dependencies.
They extend classical results to dependent, high-dimensional, U-statistic, and self-normalized contexts through advanced methods like Stein’s method and Malliavin calculus.
These bounds provide actionable insights by offering explicit error rates and constants, aiding modern probabilistic analysis and statistical inference.

A Berry–Esseen bound provides a quantitative measure of how quickly the distribution of a suitably normalized sum of random variables converges to a limiting law (typically Gaussian) in central limit theorems, quantifying this proximity via explicit, often optimal, rates in terms of moments and structural dependence. The Berry–Esseen framework has evolved from classical i.i.d. settings to encompass dependencies, functional statistics (e.g., U-statistics), operator-valued/complex random variables, multivariate and high-dimensional statistics, self-normalized forms, and non-asymptotic local limit theorems. New methods such as entropy-based arguments, concentration inequalities, exchangeable pairs, Stein’s method, and Malliavin calculus underlie modern sharp results.

1. Classical and Entropic Berry–Esseen Theorems

The classical Berry–Esseen theorem establishes that for i.i.d. real random variables $X_i$ with $\E X_i=0$, $\Var(X_i) = \sigma^2 > 0$, and $\E|X_i|^3<\infty$, the normalized sum $W_n = S_n/(\sigma\sqrt n)$ satisfies

$\sup_{x\in\mathbb{R}} |\mathbb{P}(W_n \le x) - \Phi(x)| \le C \frac{\E|X_1|^3}{\sigma^3\sqrt{n}},$

where $\Phi$ is the standard normal CDF and $C$ is an absolute constant (Zhang, 2021).

A major refinement is the entropic Berry–Esseen bound (Bobkov et al., 2011). Suppose $\{X_k\}$ are independent, centered, with variances $\sigma_k^2>0$ , and denote $\E X_i=0$0. The total variation distance between the law $\E X_i=0$1 and $\E X_i=0$2 admits

$\E X_i=0$3

where $\E X_i=0$4 (relative entropy to normality) is uniformly bounded. Relative entropy bounds (Kullback–Leibler divergence) can be derived assuming fourth moments and uniform entropy control, with $\E X_i=0$5 rates sharper than Kolmogorov-type bounds.

Table: Comparison of Classical and Entropic Berry–Esseen Rates

Setting	Metric	Moment Condition	Rate (i.i.d.)
Classical	Kolmogorov	$\E X_i=0$6	$\E X_i=0$7
Entropic	Total Variation	$\E X_i=0$8, bounded $\E X_i=0$9	$\Var(X_i) = \sigma^2 > 0$0
Entropic	Relative Entropy	$\Var(X_i) = \sigma^2 > 0$1, bounded $\Var(X_i) = \sigma^2 > 0$2	$\Var(X_i) = \sigma^2 > 0$3

The entropic approach replaces density boundedness with uniform entropy control and facilitates applications to weighted sums, $\Var(X_i) = \sigma^2 > 0$4 bounds via transportation inequalities, and explicitly exhibits limitations for discrete laws.

2. Extensions: Dependence, U-Statistics, and Graph Functionals

Berry–Esseen analysis has been extended to sums with local or graph-based dependencies (Janisch et al., 2022, Zhang, 2021, Cai et al., 2 Feb 2026, Zhang, 2021). For variables with a dependency graph $\Var(X_i) = \sigma^2 > 0$5 of degree $\Var(X_i) = \sigma^2 > 0$6,

$\Var(X_i) = \sigma^2 > 0$7

highlights explicit dependence on both the graph structure and higher moments (Janisch et al., 2022).

For generalized $\Var(X_i) = \sigma^2 > 0$8-statistics and Hoeffding decompositions, optimal $\Var(X_i) = \sigma^2 > 0$9 (or $\E|X_i|^3<\infty$0 for degenerate/edge-only kernels) rates are attainable under mild fourth-moment conditions and connectedness of principal projections (Zhang, 2021). These results leverage innovative uses of exchangeable pairs under Stein's method, accommodating complex combinatorial dependencies (e.g., subgraph counts in random graphs).

In combinatorial/probabilistic settings, e.g. Rademacher functionals or permutational statistics, modern approaches (discrete Malliavin–Stein (Krokowski et al., 2015), stratified permutation CLT (Tian et al., 18 Mar 2025)) yield Berry–Esseen rates in Kolmogorov distance that are sharp up to constants and subtle combinatorial multiplicities.

3. High-Dimensional, Multivariate, and Self-Normalized Bounds

In high-dimensional regimes, Berry–Esseen theory quantifies both the rate and the explicit dependence on ambient dimension. For sums of independent/weakly dependent $\E|X_i|^3<\infty$1-vectors $\E|X_i|^3<\infty$2 standardized such that $\E|X_i|^3<\infty$3,

$\E|X_i|^3<\infty$4

with $\E|X_i|^3<\infty$5 optimal in the purely linear case (Shao et al., 2021). For general nonlinear statistics, $\E|X_i|^3<\infty$6 dependence is currently the best known, and further refinements remain a research frontier (Shao et al., 2021). For high-dimensional martingale difference sequences, recent work establishes $\E|X_i|^3<\infty$7 rates over rectangles of $\E|X_i|^3<\infty$8 with only polylogarithmic dimension dependence, reflecting inherent limitations due to the third-moment regime (Wu et al., 4 May 2026).

Self-normalized Berry–Esseen bounds—the basis for inference using Studentized or variance-ratio statistics—are sharp: for martingale differences $\E|X_i|^3<\infty$9,

$W_n = S_n/(\sigma\sqrt n)$ 0

where $W_n = S_n/(\sigma\sqrt n)$ 1 tracks $W_n = S_n/(\sigma\sqrt n)$ 2-moments and the difference between sample and predictable quadratic variation, with $W_n = S_n/(\sigma\sqrt n)$ 3 order shown to be optimal (Fan et al., 2017, Zhang, 2021). Leung–Shao (Leung et al., 2023) provide nonuniform bounds for Studentized $W_n = S_n/(\sigma\sqrt n)$ 4-statistics, the first with valid nonuniform $W_n = S_n/(\sigma\sqrt n)$ 5 decay modulo exponentially small correction, essential for bootstrap and tail probability control.

4. Local Limit Theorems and Density-Level Berry–Esseen Bounds

For precise proximity at the density level (“local” Berry–Esseen), optimal non-asymptotic results require even sharper control. In the multivariate case, under bounded densities $W_n = S_n/(\sigma\sqrt n)$ 6, variance $W_n = S_n/(\sigma\sqrt n)$ 7,

$W_n = S_n/(\sigma\sqrt n)$ 8

where $W_n = S_n/(\sigma\sqrt n)$ 9 is the uniform directional Lyapunov third-moment, and extra $\sup_{x\in\mathbb{R}} |\mathbb{P}(W_n \le x) - \Phi(x)| \le C \frac{\E|X_1|^3}{\sigma^3\sqrt{n}},$0 reflects the smoothing cost of density-level control (Bobkov et al., 2024). If all summands are symmetric, the fourth moment $\sup_{x\in\mathbb{R}} |\mathbb{P}(W_n \le x) - \Phi(x)| \le C \frac{\E|X_1|^3}{\sigma^3\sqrt{n}},$1 replaces $\sup_{x\in\mathbb{R}} |\mathbb{P}(W_n \le x) - \Phi(x)| \le C \frac{\E|X_1|^3}{\sigma^3\sqrt{n}},$2, but at the cost of an extra $\sup_{x\in\mathbb{R}} |\mathbb{P}(W_n \le x) - \Phi(x)| \le C \frac{\E|X_1|^3}{\sigma^3\sqrt{n}},$3.

Log-concave, isotropic cases attain $\sup_{x\in\mathbb{R}} |\mathbb{P}(W_n \le x) - \Phi(x)| \le C \frac{\E|X_1|^3}{\sigma^3\sqrt{n}},$4 for densities, but in general, the rate for densities (as opposed to distribution functions) is dominated by maximal density and Lyapunov coefficients.

5. Berry–Esseen Bounds Beyond the Classical Setting: Operator-Valued, Complex, and Non-Gaussian Limits

The extension of Berry–Esseen-type estimates into operator-valued free probability, complex Wiener–Itô integrals, and non-Gaussian limiting laws is now established.

For operator-valued free CLT,

$\sup_{x\in\mathbb{R}} |\mathbb{P}(W_n \le x) - \Phi(x)| \le C \frac{\E|X_1|^3}{\sigma^3\sqrt{n}},$5

with $\sup_{x\in\mathbb{R}} |\mathbb{P}(W_n \le x) - \Phi(x)| \le C \frac{\E|X_1|^3}{\sigma^3\sqrt{n}},$6 at the resolvent/Cauchy transform level, where $\sup_{x\in\mathbb{R}} |\mathbb{P}(W_n \le x) - \Phi(x)| \le C \frac{\E|X_1|^3}{\sigma^3\sqrt{n}},$7 aggregates operator-valued moment information (Banna et al., 2021).

In the complex multiple Wiener–Itô setting, optimal Wasserstein bounds in terms of kernel contractions and fourth-moment cumulants match those in the real case, with necessary and sufficient “fourth-moment contraction” conditions identified for the complex CLT (Chen et al., 2024). Rates $\sup_{x\in\mathbb{R}} |\mathbb{P}(W_n \le x) - \Phi(x)| \le C \frac{\E|X_1|^3}{\sigma^3\sqrt{n}},$8 replace classical $\sup_{x\in\mathbb{R}} |\mathbb{P}(W_n \le x) - \Phi(x)| \le C \frac{\E|X_1|^3}{\sigma^3\sqrt{n}},$9 (as appropriate for Ornstein–Uhlenbeck processes in continuous time).

For isotonic regression and other non-regular settings, nonnormal Chernoff-type limits arise; Han–Kato derive Berry–Esseen rates matching those for oracle local average estimators, up to poly-logarithmic factors, via new anti-concentration inequalities and localization (Han et al., 2019).

6. Methodological Innovations: Stein’s Method, Concentration, Malliavin Calculus

Stein’s method, including the exchangeable pairs and zero-bias couplings, underlies most modern Berry–Esseen advances. For dependent structures, innovations include stratified permutations (Tian et al., 18 Mar 2025), concentration-of-measure via randomized windows (Cai et al., 2 Feb 2026), and stochastic calculus on discrete/Malliavin spaces (Privault et al., 2020, Krokowski et al., 2015). The same tools are fundamental to both the classical bounds and extensions, with moment and contraction terms encoding the relevant combinatorics and dependence.

Non-asymptotic control—even in complex dependency structures, functional statistics (e.g., entropy estimators (Yu et al., 13 Apr 2026)), or operator-algebraic regimes—is now available, with explicit Berry–Esseen constants and rates in most contemporary inferential settings.

7. Summary Table: Berry–Esseen Bounds in Diverse Settings

Structure	Metric	Rate (sharpest)	Key Condition(s)	Reference
I.I.D., sum	Kolmogorov	$\Phi$ 0	$\Phi$ 1	(Zhang, 2021)
Non-i.i.d., entropy controlled	TV/entropy	$\Phi$ 2, $\Phi$ 3	$\Phi$ 4/ $\Phi$ 5, $\Phi$ 6	(Bobkov et al., 2011)
Sparse dependency graph	Kolmogorov	$\Phi$ 7	bounded third ( $\Phi$ 8)	(Janisch et al., 2022)
$\Phi$ 9-statistics	Kolmogorov	$C$ 0	$C$ 1<∞, connectedness	(Zhang, 2021)
High-d martingale diff.	Kolmogorov (rect)	$C$ 2	thrid moments, cond. var.	(Wu et al., 4 May 2026)
Self-normalized sums	Kolmogorov	$C$ 3	$C$ 4th moments, local dep.	(Zhang, 2021, Cai et al., 2 Feb 2026)
Permutation/statistical design	Kolmogorov	$C$ 5	$C$ 6rd moments	(Tian et al., 18 Mar 2025)
Complex/operator-valued	Wasserstein/Levy	$C$ 7 (resolvent), $C$ 8 (prob.)	moments, kernel contractions	(Banna et al., 2021, Chen et al., 2024)
Isotonic regression	Non-Gaussian	$C$ 9 (Chernoff)	model smoothness	(Han et al., 2019)
Plug-in entropy/diversity	Kolmogorov	$\{X_k\}$ 0	H\"older regularity, tail	(Yu et al., 13 Apr 2026)

All rates are up to constants; actual rates and dependencies on structure-specific combinatorial or analytic parameters are explicit in the cited references.

References:

(Bobkov et al., 2011) Bobkov–Chistyakov–Götze, "Berry–Esseen bounds in the entropic central limit theorem"
(Zhang, 2021) Shao–Zhang, "Berry--Esseen bounds for generalized $\{X_k\}$ 1 statistics"
(Janisch et al., 2022) Janisch–Le Héricy, "Berry-Esseen-type estimates for random variables with a sparse dependency graph"
(Cai et al., 2 Feb 2026) Cai–Shao–Zhang, "Refined Berry-Esseen bounds under local dependence"
(Wu et al., 4 May 2026) Shao–Zhang, "Berry-Esseen bounds for multivariate martingale difference sequences in the Kolmogorov distance"
(Zhang, 2021) Shao–Zhang, "Berry--Esseen bounds for self-normalized sums of local dependent random variables"
(Krokowski et al., 2015) Krokowski–Reichenbachs–Thäle, "Discrete Malliavin–Stein method: Berry-Esseen bounds for random graphs and percolation"
(Tian et al., 18 Mar 2025) "Stratified Permutational Berry--Esseen Bounds and Their Applications to Statistics"
(Bobkov et al., 2024) Bobkov–Götze, "Berry-Esseen bounds in local limit theorems"
(Privault et al., 2020) Privault–Serafin, "Berry-Esseen bounds for functionals of independent random variables"
(Chen et al., 2024) "Berry-Esséen bound for complex Wiener-Itô integral"
(Banna et al., 2021) "Berry-Esseen bounds for the multivariate $\{X_k\}$ 2-free CLT and operator-valued matrices"
(Han et al., 2019) Han–Kato, "Berry-Esseen bounds for Chernoff-type non-standard asymptotics in isotonic regression"
(Yu et al., 13 Apr 2026) Yu–Miao, "Berry-Esseen bounds for estimators of entropy and diversity indices on countable alphabets"
(Fan et al., 2017) Fan–Shao, "Berry-Esseen bounds for self-normalized martingales"
(Leung et al., 2023) Leung–Shao, "Nonuniform Berry-Esseen bounds for Studentized U-statistics"
(Shao et al., 2017) Shao–Zhang, "Berry-Esseen Bounds of Normal and Non-normal Approximation for Unbounded Exchangeable Pairs"