Non-Asymptotic Berry–Esseen Bounds

• Non-Asymptotic Berry–Esseen Bounds are explicit finite-sample estimates that quantify the error in approximating a normalized statistic by a normal law.
• They employ advanced methods such as Stein’s method, Malliavin calculus, and recursive reductions to handle dependencies and structure in data.
• Applications include random graphs, U-statistics, self-normalized sums, and random matrices, thereby providing practical assessments in modern probabilistic analysis.

Non-Asymptotic Berry--Esseen Bounds provide explicit, finite-sample quantitative estimates for the approximation error in central limit theorems (CLT), measured typically via the Kolmogorov or total variation distance between a normalized statistic and the standard normal law, without requiring $n \to \infty$ asymptotics. These bounds are indispensable in modern probability, statistics, and combinatorics for quantifying the accuracy of normal approximations in settings featuring dependency, structure, or non-i.i.d. data, with applications ranging from random graphs, U-statistics, dynamical systems, and random matrices, to self-normalized sums and non-classical CLT limits.

1. Fundamental Concepts and General Framework

A non-asymptotic Berry--Esseen bound typically asserts, for a normalized sum or functional $W_n$, that

$$\sup_{z \in \mathbb{R}} |P(W_n \leq z) - \Phi(z)| \leq \frac{C}{r_n}$$

where $\Phi(z)$ is the standard normal cdf, $C$ is a universal or explicit constant independent of $n$ (and other parameters), and $r_n$ is a quantity (usually diverging with $n$ or the sample size/variance) depending on the statistic or model structure. Unlike classical asymptotic statements, these results provide finite-sample error rates with all dependencies made explicit.

Crucial to these results, particularly in dependent or structured settings, are alternative techniques beyond direct characteristic function manipulation. These include:

• Stein's method, often implemented via exchangeable pairs, size bias, or Malliavin calculus.
• Inductive and recursive decompositions, allowing complex combinatorial or dependent structures to be reduced and controlled recursively.
• Coupling and difference operators, to manage the deviation between the statistic of interest and its reference limit.

2. Main Methodologies

Stein's Method and Non-Uniform Couplings

Stein's method generates non-asymptotic bounds by relating target distributions to the normal law via the solution to the Stein equation: $f'(w) - w f(w) = h(w) - \mathbb{E}[h(Z)]$ for smooth $h$. In classical normal approximation, bounds on $\mathbb{E} f'(W) - \mathbb{E} Wf(W)$ are directly translated to Kolmogorov or total variation distances.

A technical obstacle is that, in settings with dependencies or lack of bounded increments (e.g., random graphs, unbounded couplings), classical exchangeable pairs or size-bias techniques are insufficient. Recent advances remove uniform boundedness requirements—for example, when the difference $|Y^s_n - Y_n|$ can be large, error control is achieved via moment conditions: $$\mathbb{E}\left[(1 + |W_n|) (Y^s_n - Y_n)^2\right] < \infty$$ enabling the use of size bias coupling in unbounded, combinatorial contexts (Goldstein, 2010).

Recursive and Inductive Reduction

Especially in structured models such as random graphs, recursive arguments allow bounding the error for $n$-vertex objects via those for $n-1$ or $n-\ell$ objects by explicitly relating functionals on the full structure to those on subsamples (e.g., removing a randomly chosen vertex and adjusting connectivity) (Goldstein, 2010).

The Berry--Esseen error term is shown to satisfy a recursion of the form: $a_n \leq \sum_{\ell} a_{n-\ell} p_{n,\ell} + f$ which, subject to boundedness of increments and control of coefficients, yields uniform non-asymptotic bounds.

Malliavin Calculus and Discrete Malliavin--Stein

For functionals expressible as sums of chaos (i.e., polynomials of independent variables, U-statistics, counts in random geometric structures), discrete or Gaussian Malliavin calculus combined with Stein's method gives quantitative control via moments of discrete gradients and contractions (Krokowski et al., 2015, Privault et al., 2020, Lachièze-Rey et al., 2015, Nourdin et al., 2018).

The key feature is direct control over the variance and higher moments through so-called difference or finite-difference operators: $\Delta_j f(X, X') := f(X) - f(X^{\{j\}}, X')$ whose moments (particularly 3rd, 4th, and 6th order) control the error in Kolmogorov or Wasserstein distances.

Fourier and Cumulant Methods for Dependent Structures

For sums of dependent variables organized via a graphical structure (e.g., dependency graph), Fourier-analytic techniques provide Berry--Esseen-type bounds with explicit correction factors in terms of the maximum degree or local dependency measure (Janisch et al., 2022). The Berry--Esseen error then scales with the "effective" independence, decaying as the dependency graph stays sparse.

3. Key Examples and Explicit Results

Random Graphs and Combinatorial Models

In the context of the Erdős-Rényi random graph $G_n$, let $Y_n$ denote the number of vertices of given degree $d$. Setting

$$W_{n,\theta} = \frac{Y_n - \mu_{n,\theta}}{\sigma_{n,\theta}}$$

with explicit $\mu_{n, \theta} = n \tau_{n,\theta}$ and

$$\sigma^2_{n,\theta} = n\tau_{n,\theta}^2\left(\frac{(d - \theta)^2}{\theta (1 - \theta/(n-1))} - 1\right) + n\tau_{n,\theta}$$

and

$$\tau_{n, \theta} = \binom{n-1}{d} (\theta/(n-1))^d (1 - \theta/(n-1))^{n-1-d}$$

the explicit non-asymptotic Berry--Esseen bound is

$$\sup_{z \in \mathbb{R}} |P_{\theta}(W_{n, \theta} \leq z) - P(Z \leq z)| \leq \frac{C}{\sqrt{n\tau_\theta}} \quad\text{where}\quad \tau_\theta = e^{-\theta} \theta^d/d!$$

which holds uniformly for finite $n \geq d + 1$ and all $\theta \in (0, b]$ (Goldstein, 2010).

Self-Normalized and Martingale Sums

For self-normalized sums $T = S/V$ with $S = \sum_{i} X_i$, $V = (\sum_i X_i^2)^{1/2}$, explicit non-asymptotic bounds involving higher moments are proved: $$|P(T \leq z) - \Phi(z)| \leq A_3 \frac{\Delta_3}{\Delta_2^{3/2}} + A_4 \frac{\Delta_4^{1/2}}{\Delta_2} + A_6 \frac{\Delta_6}{\Delta_3^3\Delta_2^{3/2}}$$ for all $z \in \mathbb{R}$, where $\Delta_p = \sum_i |X_i|^p$ and the constants $A_3,A_4,A_6$ are explicitly listed (Pinelis, 2011), leading to practical assessments in t-tests and other statistical applications.

For martingale difference arrays, the self-normalized Berry--Esseen bound is

$$\sup_x |P(S_n/[S]_n \leq x) - \Phi(x)| \leq C_p N_n^{1/(2p+1)}$$

where $N_n$ involves the $2p$-th moments and the deviation of conditional variance from unity (Fan et al., 2017).

Dependency Graphs

Given a triangular array $\{Y_{n,k}\}$ with bounded $L^\delta$-norms and dependency graph of degree $D(n)$,

$$d_{\text{Kol}}\left(\frac{S_n - \mathbb{E}[S_n]}{\sqrt{\mathrm{Var}[S_n]}}, \mathcal{N}(0,1)\right) \leq \max\left\{ C_1 \frac{(D+1)^2 \mathcal{A}_3}{v^3}, C_2 \frac{L(D+1)}{v} \right\}$$

with explicit formulas for all parameters, so as $D \ll N(n)$, convergence remains nearly as sharp as the i.i.d. case (Janisch et al., 2022).

Higher-Order and Smoothness-Enhanced Bounds

If the first $k \geq 3$ moments of $X$ match the standard normal, $X$ has finite $(k+1)$st moment, and $X$ has density at least $h$ on an interval of width $w$,

$$\sup_{s \in \mathbb{R}} \left| P\left( \frac{X_1 + \cdots + X_N}{\sqrt{N}} \leq s \right) - \Phi(s) \right| \leq C(k) \frac{\mathbb{E}|X|^{k+1}}{N^{(k-1)/2}} + 3\exp\left(-c h w^3 N / \mathbb{E}|X|^{k+1}\right)$$

so, for $k=3$, symmetric distributions with densities and finite fourth moment satisfy a Berry--Esseen inequality of order $1/N$, improving upon the classical $1/\sqrt{N}$ rate under minimal additional smoothness (Johnston, 2023).

4. Applications Across Domains

Non-asymptotic Berry--Esseen bounds have enabled the following advances:

• Random graphs: Distributional limits for subgraph and vertex degree counts with explicit error rates (Goldstein, 2010, Krokowski et al., 2015).
• U-statistics and incomplete U-statistics: Validity of normal approximations for nonlinear functionals under minimalistic moment assumptions, including Bernoulli sampling and high-dimensional regimes (Leung, 8 Jun 2024, Privault et al., 2020, Leung et al., 2023).
• Random matrices and free probability: Operator-valued settings with explicit Lévý and Kolmogorov distance rates for joint semicircular limits and polynomial test functions (Banna et al., 2021).
• Martingale theory: Precise bounds matching the best possible rates for self-normalized martingales and least-squares estimators in time series (Fan et al., 2017).
• Small deviations and moment methods: Hybrid approaches combining Berry--Esseen bounds and SDP-moment methods to sharply bound probabilities of rare events (e.g., Feige's conjecture) (Guo et al., 2020).
• Non-regular statistical estimation: Berry--Esseen bounds for Chernoff-type non-Gaussian, non-$\sqrt{n}$ CLT limits, notably in isotonic regression, using localization and anti-concentration tools (Han et al., 2019).

5. Limitations, Optimality, and Open Directions

• Sharpness and Optimality: For certain discrete or nearly singular distributions (e.g., Bernoulli), the $1/\sqrt{n}$ rate is optimal unless additional smoothness is imposed (Zolotukhin et al., 2018, Johnston, 2023). The presence of continuous density, even on a small interval, can yield faster convergence ($1/N$) provided matching moments.
• Model Complexity: When dependency, non-stationarity, or joint non-commutative structure is present, bounds often carry dimension-dependent constants, and polynomial or logarithmic correction factors reflecting the combinatorial intricacies.
• Regularity of Test Functions: In high-dimensional or non-smooth settings (e.g., convex set metrics in the multivariate CLT), smoothing approximations and solutions to the Stein equation require new techniques, as direct application may not be possible (Leppänen, 25 Mar 2024).
• Self-Normalization and Studentization: For statistics with random normalizers, tailor-made exponential tail inequalities and variable censoring methods are crucial to control denominator fluctuations (Leung et al., 2023, Leung, 8 Jun 2024).

Open challenges remain in determining the best possible constants (see (Zolotukhin et al., 2018) for computational progress in the Bernoulli case), extending bounds to complex dependence networks, and integrating these methods into high-dimensional and non-classical CLT frameworks.

6. Tabular Overview of Representative Results

Paper/Setting	Statistic/Model	Berry–Esseen Rate
(Goldstein, 2010) Random graphs (vertex degrees)	Count of degree $d$ vertices	$C/\sqrt{n e^{-\theta}\theta^d/d!}$
(Krokowski et al., 2015) Triangle counts in $G(n,p)$	Normalized triangle count	$O(n^{-1+\alpha})$ ($p = \theta n^{-\alpha}$)
(Pinelis, 2011) Student/self-normalized sums	$T = S/V$	$$O(\Delta_3/\Delta_2^{3/2} + \Delta_4^{1/2}/\Delta_2)$$
(Johnston, 2023) High-moment, density-matching $X$	$\sum X_i/\sqrt{N}$, first $k$ moments matched	$O(1/N)$ (for $k=3$ and extra density)
(Janisch et al., 2022) Dependency graph (max degree $D$)	Sums with dependency graph	$O((D+1)^2/\sqrt{N})$

7. Conclusion

Non-Asymptotic Berry--Esseen Bounds constitute a pivotal set of results that both sharpen and extend the classical theory of normal approximation. By leveraging advanced techniques—Stein's method with sophisticated couplings, Malliavin calculus, inductive schemes, Fourier analysis, and computational optimization—they deliver explicit, finite-sample rates for a broad range of statistics, often under minimal assumptions and in highly structured or dependent models. This machinery is central to modern probabilistic analysis, risk quantification, and statistical inference especially where asymptotic statements are insufficient or inapplicable.