Non-uniform Berry–Esseen Bounds

• The non-uniform Berry–Esseen bound refines classical CLT rates by providing pointwise error estimates with explicit dependence on the approximation point z.
• It employs techniques like Fourier smoothing, Stein’s method, and Malliavin calculus to achieve precise control over moderate deviations and tail probabilities.
• These bounds are crucial for accurate rare-event probability estimation and statistical inference in settings such as dependent, martingale, and self-normalized sequences.

A non-uniform Berry–Esseen (BE) bound is a refinement of classical central limit theorem (CLT) rates, providing pointwise bounds on the Kolmogorov distance between the distribution of a normalized sum or functional and the normal law, with an explicit dependence on the approximation point $z$ rather than only a global supremum. Such bounds take the form $|\mathbb{P}(T \le z)-\Phi(z)| \le$ (error term decaying in $z$), and are crucial for precise quantitative control in the moderate and large deviations regimes, especially for rare-event probability estimation and statistical inference involving extreme quantiles. The literature provides a spectrum of techniques—Fourier smoothing, Stein's method, Malliavin calculus, and concentration inequalities—producing a variety of non-uniform BE bounds across classical, dependent, functional, martingale, and self-normalized settings.

1. Fundamental Forms and Historical Milestones

The archetype for non-uniform BE bounds arises from sums of independent real random variables $X_1,\dots,X_n$ with $E[X_i]=0$, $B^2 = \sum_{i=1}^n \operatorname{Var}(X_i)$, and $A = \sum_{i=1}^n E|X_i|^3<\infty$. The original bound of Nagaev (1965/1976) and subsequent sharp versions state

$$|P(S/B>z) - P(Z>z)| \le C_{\mathrm{nu}} \frac{A}{B^3} \frac{1}{1+z^3}, \qquad z \ge 0$$

where $S = \sum X_i$ and $Z\sim N(0,1)$. The uniform (classical) BE bound lacks the explicit $|\mathbb{P}(T \le z)-\Phi(z)| \le$0-dependence and is typically, for iid, $|\mathbb{P}(T \le z)-\Phi(z)| \le$1. The best-known uniform constant $|\mathbb{P}(T \le z)-\Phi(z)| \le$2 is within 15% of optimality, but $|\mathbb{P}(T \le z)-\Phi(z)| \le$3 has only recently been reduced to $|\mathbb{P}(T \le z)-\Phi(z)| \le$4 (Pinelis, 2013).

Advancements include:

• Smoothing inequalities delivering optimal constants (Pinelis, 2013, Pinelis, 2013).
• Non-uniform bounds for functionals (chaos, martingales, Poisson, self-normalized, exchangeable pair regimes).
• Replacement of polynomial decay $|\mathbb{P}(T \le z)-\Phi(z)| \le$5 by exponential $|\mathbb{P}(T \le z)-\Phi(z)| \le$6 as achievable under additional tail or moment conditions.

2. Methodological Paradigms

2.1 Fourier Analytic Smoothing and Truncation

The classical approach by Nagaev involves a blend of truncation, Cramér–Esséen tilting for moderate deviations, and exponential tail inequalities beyond the uniform-case range. The introduction of smoothing inequalities (cf. Prawitz filter, symmetric kernels) enables direct passage from characteristic function bounds to non-uniform error estimates, bypassing delicate explicit density/Edgeworth expansions, and yielding improvements such as

$|\mathbb{P}(T \le z)-\Phi(z)| \le$7

with $|\mathbb{P}(T \le z)-\Phi(z)| \le$8 as small as $|\mathbb{P}(T \le z)-\Phi(z)| \le$9–$z$0 for suitable filtering and numerical optimization (Pinelis, 2013, Pinelis, 2013).

2.2 Stein's Method and Concentration

Stein's method, when coupled with concentration inequalities, provides a flexible framework for both normal and non-normal approximation in classical and dependent settings. The central technical element is the construction and control of the solution $z$1 to the Stein equation,

$z$2

with explicit bounds on $z$3, $z$4, and higher derivatives allowing one to obtain non-uniform Kolmogorov error bounds via tailored moment and concentration controls (Pinelis, 2011, Thành et al., 15 Feb 2025, Thành et al., 20 Jun 2025).

2.3 Malliavin–Stein Theory

In functionals of Gaussian, Poisson, or Rademacher processes (e.g., multiple Wiener–Itô integrals), the Malliavin–Stein approach gives non-uniform BE bounds depending on Malliavin-derivative-based quantities such as

$z$5

and controls the error via

$z$6

with $z$7 in an appropriate Sobolev space $z$8 (Dung et al., 2024, Butzek et al., 2024).

3. Key Theorems and Typical Non-uniform Bounds

3.1 Sums of Independent Variables

Setting	Pointwise Bound	Reference
Classical iid sum	$z$9	(Pinelis, 2013, Pinelis, 2013)
Exponential decay (3rd mom)	$X_1,\dots,X_n$0	(Pinelis, 2011)
Higher moments $X_1,\dots,X_n$1	$X_1,\dots,X_n$2	(Pinelis, 2011)

3.2 Exchangeable Pairs

For an exchangeable pair $X_1,\dots,X_n$3 with

$X_1,\dots,X_n$4

and under finite $X_1,\dots,X_n$5-moment, unbounded $X_1,\dots,X_n$6, the general non-uniform BE bound reads

$X_1,\dots,X_n$7

with $X_1,\dots,X_n$8 collecting $X_1,\dots,X_n$9 deviations from ideal conditional variance, regression remainders, and large/small jump controls (Thành et al., 15 Feb 2025, Liu et al., 2019).

3.3 Functionals of Gaussian and Poisson Processes

For $E[X_i]=0$0, $E[X_i]=0$1, $E[X_i]=0$2,

$E[X_i]=0$3

with $E[X_i]=0$4. For multiple Wiener–Itô integrals $E[X_i]=0$5, $E[X_i]=0$6 and explicit exponential tail controls are available for $E[X_i]=0$7 (Butzek et al., 2024, Dung et al., 2024).

3.4 Martingales

For a martingale difference array satisfying the conditional Bernstein condition,

$E[X_i]=0$8

with model-specific expressions for $E[X_i]=0$9 in terms of $B^2 = \sum_{i=1}^n \operatorname{Var}(X_i)$0 controlling tail and quadratic variation deviations (Fan et al., 2016, Wu et al., 2021).

4. Applications and Model-Specific Instantiations

• Multiple Wiener–Itô integrals: Non-uniform bounds decay exponentially in $B^2 = \sum_{i=1}^n \operatorname{Var}(X_i)$1, critical for CLTs in fixed Wiener chaos (Dung et al., 2024, Butzek et al., 2024).
• Exponential functionals of Brownian motion: Quantitative non-uniform bounds track the deviation of the normalized functional from normality in settings with heavy tailed or log-normal upper tails (Dung et al., 2024).
• Spin Model Macroscopic Observables: In mean-field Curie–Weiss and $B^2 = \sum_{i=1}^n \operatorname{Var}(X_i)$2-vector models, non-uniform bounds for magnetization or squared spin length achieve optimal $B^2 = \sum_{i=1}^n \operatorname{Var}(X_i)$3 uniform rates with an additional decay in $B^2 = \sum_{i=1}^n \operatorname{Var}(X_i)$4 (Thành et al., 15 Feb 2025, Thành et al., 20 Jun 2025).
• Martingale Regression and Self-normalization: Adapted regression statistics and self-normalized CLTs inherit polynomial or exponential tail decay via non-uniform BE inequalities, sidestepping strong moment or independence assumptions (Fan et al., 2016, Wu et al., 2021).
• Studentized U-Statistics: Necessary correction terms of exponential type are introduced to avoid failures in naive non-uniform bounds due to vanishing denominators, ensuring valid rates for $B^2 = \sum_{i=1}^n \operatorname{Var}(X_i)$5-statistic and higher-degree kernels (Leung et al., 2023).
• Weakly Dependent Sequences: Stationary Markov chains, dynamical systems, and products of random matrices allow non-uniform BE and Edgeworth expansion with rate $B^2 = \sum_{i=1}^n \operatorname{Var}(X_i)$6 under general cumulant-derivative controls (Hafouta, 2022).

5. Technical Proof Elements and Optimality

5.1 Stein–Malliavin Integration by Parts

In Gaussian and Poisson functionals, the key representation

$B^2 = \sum_{i=1}^n \operatorname{Var}(X_i)$7

enables a split of the error over small and large $B^2 = \sum_{i=1}^n \operatorname{Var}(X_i)$8 via tailored bounds for $B^2 = \sum_{i=1}^n \operatorname{Var}(X_i)$9 and $A = \sum_{i=1}^n E|X_i|^3<\infty$0. The Cauchy–Schwarz step and concentration inequalities enable exponential or polynomial tail decay, with constants coming from precise operator norms or Malliavin derivatives (Dung et al., 2024, Butzek et al., 2024).

5.2 Moment Truncation and Smoothing

Smoothing inequalities using compactly supported filters applied to the characteristic function produce directly the necessary $A = \sum_{i=1}^n E|X_i|^3<\infty$1 factor and uniform constants, as in

$A = \sum_{i=1}^n E|X_i|^3<\infty$2

with further refinements tied to the chosen filter, truncation scales, and analysis of remainder terms in the Edgeworth expansion (Pinelis, 2013, Pinelis, 2013).

6. Optimality, Limitations, and Extensions

• The $A = \sum_{i=1}^n E|X_i|^3<\infty$3 prefactor is optimal whenever only a finite $A = \sum_{i=1}^n E|X_i|^3<\infty$4th moment is assumed for $A = \sum_{i=1}^n E|X_i|^3<\infty$5, and cannot in general be replaced by exponential decay unless stronger sub-Gaussian cores or concentration hold (Butzek et al., 2024).
• In functionals of Poisson and Rademacher variables, second-order Poincaré inequalities enable more granular control involving Malliavin operators up to order 2, permitting explicit decomposition of the error into leading and higher-order components (Butzek et al., 2024).
• For Studentized statistics, naive non-uniform bounds may fail due to rare events where the denominator is vanishingly small. Here, adding a (sharp, necessary) exponentially small correction term restores validity (Leung et al., 2023).
• In exchangeable-pair-based normal approximation, unbounded difference settings are now included with explicit control on large jumps, polynomial tail factors, and remainders (Thành et al., 15 Feb 2025, Liu et al., 2019).

7. Summary Tables: Representative Non-uniform Berry–Esseen Bounds

Class	Error Form	Decay in $A = \sum_{i=1}^n E|X_i|^3<\infty$6	Key Condition	Reference
iid sum	$A = \sum_{i=1}^n E|X_i|^3<\infty$7	Polynomial	$A = \sum_{i=1}^n E|X_i|^3<\infty$8	(Pinelis, 2013)
Sums (exp decay)	$A = \sum_{i=1}^n E|X_i|^3<\infty$9	Exponential	$$|P(S/B>z) - P(Z>z)| \le C_{\mathrm{nu}} \frac{A}{B^3} \frac{1}{1+z^3}, \qquad z \ge 0$$0	(Pinelis, 2011)
Martingale	$$|P(S/B>z) - P(Z>z)| \le C_{\mathrm{nu}} \frac{A}{B^3} \frac{1}{1+z^3}, \qquad z \ge 0$$1	Exponential	Conditional Bernstein condition	(Fan et al., 2016)
Functionals (Gaussian/Poisson)	$$|P(S/B>z) - P(Z>z)| \le C_{\mathrm{nu}} \frac{A}{B^3} \frac{1}{1+z^3}, \qquad z \ge 0$$2	Polynomial	$$|P(S/B>z) - P(Z>z)| \le C_{\mathrm{nu}} \frac{A}{B^3} \frac{1}{1+z^3}, \qquad z \ge 0$$3, $$|P(S/B>z) - P(Z>z)| \le C_{\mathrm{nu}} \frac{A}{B^3} \frac{1}{1+z^3}, \qquad z \ge 0$$4	(Butzek et al., 2024)
Exchangeable pair	$$|P(S/B>z) - P(Z>z)| \le C_{\mathrm{nu}} \frac{A}{B^3} \frac{1}{1+z^3}, \qquad z \ge 0$$5	Polynomial	$$|P(S/B>z) - P(Z>z)| \le C_{\mathrm{nu}} \frac{A}{B^3} \frac{1}{1+z^3}, \qquad z \ge 0$$6 finite	(Thành et al., 15 Feb 2025)

• "More on the nonuniform Berry--Esseen bound" (Pinelis, 2013)
• "On the nonuniform Berry--Esseen bound" (Pinelis, 2013)
• "Non-uniform Berry-Esseen bounds via Malliavin-Stein method" (Dung et al., 2024)
• "Non-uniform Berry--Esseen bounds for Gaussian, Poisson and Rademacher processes" (Butzek et al., 2024)
• "Improved nonuniform Berry--Esseen-type bounds" (Pinelis, 2011)
• "Non-uniform Berry--Esseen bounds for exchangeable pairs with applications..." (Thành et al., 15 Feb 2025)
• "Non-uniform Berry-Esseen Bound by Unbounded Exchangeable Pair Approach" (Liu et al., 2019)
• "Nonuniform Berry-Esseen bounds for Studentized U-statistics" (Leung et al., 2023)
• "Nonuniform Berry-Esseen bounds for martingales..." (Fan et al., 2016)
• "Non-uniform Berry-Esseen theorem and Edgeworth expansions..." (Hafouta, 2022)
• "Nonuniform Berry-Esseen bound for self-normalized martingales" (Wu et al., 2021)
• "Non-uniform bounds for non-normal approximation via Stein's method..." (Thành et al., 20 Jun 2025)