Berry–Esseen Rates in CLT Convergence

• Berry–Esseen type rates are explicit bounds that quantify the speed at which normalized sums converge to their limiting distributions in CLTs.
• They depend on moment conditions and dependency structures, varying across independent, dependent, and non-commutative frameworks to determine optimal convergence rates.
• Applications include refining quantitative approximations in random matrix theory, heavy-tailed processes, and functional analysis within probability.

Berry–Esseen type rates quantify convergence speeds in central limit theorems (CLTs) by giving explicit upper bounds for the distance between normalized sums (or, more generally, functionals of random objects) and their limiting distributions, often Gaussian. The rates depend on moment assumptions, dependence structure, group/action context, and the underlying probability operations (classical, free, Boolean, monotone, etc.). Berry–Esseen bounds play a central role in quantitative limit theory and its applications.

1. Classical Berry–Esseen Rate and Generalizations

The classical Berry–Esseen theorem asserts, for sums $S_n = X_1 + \cdots + X_n$ of independent, mean-zero, variance-one random variables with $\mathbb{E}|X_1|^3<\infty$,

$$\sup_x \left| \mathbb{P}\left( \frac{S_n}{\sqrt{n}} \le x \right) - \Phi(x) \right| \le C \frac{\mathbb{E}|X_1|^3}{\sqrt{n}}$$

where $\Phi$ is the standard normal cdf. The rate $O(n^{-1/2})$ with leading constant (typically $C \le 0.56$) is optimal as long as the third moment is finite (Döbler, 2012). Analogous rates in other metrics (Kolmogorov, total variation, Wasserstein) require corresponding moment and regularity assumptions (Bobkov et al., 2011).

2. Berry–Esseen Bounds under Weak and Dependent Structures

For weakly dependent stationary processes $X_k = g_k(\epsilon_k, \epsilon_{k-1}, \ldots)$ with dependence measured via coupling ($$\delta_p(\ell) = \sup_k \| X_k - X_k^{(\ell,\,\prime)} \|_p$$), the Berry–Esseen rate is $O(n^{p/2-1})$ for $2 < p \le 3$ under minimal summability $\sum_\ell \ell^2 \delta_p(\ell) < \infty$ (Jirak, 2016). For martingale difference arrays, Bolthausen’s 1982 result and its extensions yield

$$\sup_x \left| \mathbb{P}(M_n/\sqrt{n} \le x ) - \Phi(x) \right| = O(n^{-1/4})$$

for bounded fourth moments. However, for randomized martingale projections (random spherical weights), faster rates $O(n^{-1})$ are achievable under $L^4$–bounds, leveraging the principle of conditioning and spherical concentration (Dedecker et al., 12 May 2025).

Dependent graphs with sparse connectivity ($\Delta = \max$ degree) allow bounds of the form

$$d_K\left( \frac{S - \mathbb{E}S}{\sigma}, \mathcal N(0,1) \right) = O\left( \left( \frac{\Delta+1}{n} \right)^{(\delta-2)/2(\delta+1)} \right)$$

for random variables with uniformly bounded $\delta$-moments, further extending the rates to $m$-dependent and local dependence situations (Janisch et al., 2022).

3. Matrix and Group-valued Random Products: GL$_d(\mathbb{R})$ and Beyond

In random walk dynamics on $GL_d(\mathbb{R})$, convergence rates for $\log \| A_n \|$ (where $A_n$ is the product of $n$ i.i.d. matrices) depend strongly on moment conditions and irreducibility/proximality. If $\mu$ has a polynomial moment of order $q \in (2,3]$,

$$\sup_x \left| F_n(x) - \Phi(x) \right| \le C \left( \frac{\log n}{n} \right)^{q/2-1}$$

whereas with a fourth moment, the optimal $O(n^{-1/2})$ rate is attained (Cuny et al., 2022). More refined multi-dimensional, coefficient, and spectral radius bounds require smoothing lemmas and spectral gap techniques (Cuny et al., 2021).

4. Non-Commutative Probabilities: Free, Boolean, Monotone Convolution

In free probability, the Berry–Esseen-type estimate in the free CLT under finite fourth moment is

$$\Delta(\mu_{S_n}, w) \le C_\varepsilon (L_{4,n})^{1/2-\varepsilon/2}$$

where $L_{4,n}$ is the normalized fourth moment, and $w$ is the semicircle law; for i.i.d. case, the rate is "almost" $n^{-1/2}$ [$2503.23403$]. Finite free analogues achieve an exact $O(n^{-1/2})$ in Lévy distance between empirical root measures (Arizmendi et al., 2023).

Boolean convolution central limit theorems yield, for a probability measure $\mu$ with finite fourth moment,

$$L\Big(D_{1/\sqrt{n}} (\mu^{\uplus n}),\ b\Big) \le 7 \sqrt{ \frac{m_4(\mu)-1}{n} }$$

with limit $b = (1/2)\delta_{-1} + (1/2)\delta_{1}$ and $L$ the Lévy distance (Arizmendi et al., 2017). Kolmogorov distance is not suitable due to the nature of the limit; optimality of $\sqrt{1/n}$ holds in this setting.

Monotone convolution CLTs have strictly slower rates: under finite fourth moment, the distance between normalized monotone convolution powers and the arcsine law is $O(n^{-1/8})$, with possible improvement to $O(n^{-1/4})$ given higher regularity. This non-commutative setting reflects more delicate analytic growth (Arizmendi et al., 2018).

5. Berry–Esseen Bounds in Multivariate and Entropic Local Limit Theorems

Multivariate local limit theorems offer explicit Berry–Esseen–type bounds on the supremum norm of the difference between densities of normalized sums and the Gaussian: $$\sup_{x \in \mathbb{R}^d} |p_n(x) - \varphi(x)| \le C^d \sigma^d M^2 \frac{B_3}{\sqrt{n}}$$ where $M$ bounds densities, and $B_3$ is the Lyapunov coefficient; symmetry or vanishing third cumulant sharpens this to $O(1/n)$. Rates depend exponentially on dimension, with log-concavity and product structure taming the curse of dimensionality (Bobkov et al., 2024).

Entropic Berry–Esseen theorems yield $O(L_3)$ bounds in total variation, and $O(L_4)$ in relative entropy, where $L_3$ and $L_4$ are Lyapunov ratios; optimal rates are $O(n^{-1/2})$ and $O(n^{-1})$, respectively, in i.i.d. settings (Bobkov et al., 2011).

6. Rates for Heavy-tailed, Regenerative, and Functional Processes

For heavy-tailed moving averages with kernel parameter $\alpha$ and tail index $\beta$ (Lévy density $\sim |x|^{-1-\beta}$), the Berry–Esseen rate is

$$d_W(S_n, Z) \le C n^{-1/2} \quad \text{if } \alpha\beta > 3, \qquad d_K(S_n, Z) \le C n^{-1/2} \quad \text{if } \alpha\beta > 4$$

but deteriorates below critical thresholds as joint heavy tails and long memory interact (Basse-O'Connor et al., 2019).

Regenerative processes under a $(2+\delta)$th moment hypothesis yield

$$\sup_x | \mathbb{P} \{ (X_n - n\mu)/(\sigma \sqrt{n}) \le x \} - \Phi(x) | \le C n^{-\delta/2}$$

where $\delta \in (0,1]$ depending on moment assumptions; classical $O(n^{-1/2})$ is recovered when $\delta=1$ (Guo et al., 2017).

Functional settings such as the Wiener chaos admit rates governed sharply by third and fourth cumulants: if $F_n$ is normalized in a fixed chaos,

$$d(F_n, N) \asymp \max(|\mathbb{E}[F_n^3]|, \mathbb{E}[F_n^4] - 3)$$

and no further improvement is possible without additional structure (Biermé et al., 2011).

7. Optimality, Smooth Test Functions, and Special Metrics

Optimal Berry–Esseen bounds for smooth test functions $f \in C^{2,1}$ involve strictly sharper constants: $$\left| \mathbb{E} f\left(\frac{S_n-\sum \mu_i}{\sigma}\right) - \mathbb{E}f\left(\frac{T_n}{\sigma}\right) \right| \le \frac{L}{6} \sum_{i=1}^n \frac{\sigma_i^3}{\sigma^3} B(\rho_i)$$ where $L$ bounds the Lipschitz norm of $f''$, $T_n$ the sum of isoscedastic symmetrized two-point variables, and $B(\rho)$ an optimal concave growth term in the standardized third moment. This refines classical results (e.g., Tyurin), yielding multiplicative improvements, especially in homoscedastic and i.i.d. settings (Mattner et al., 2017).

For the $\chi^2$ metric, higher moment matching accelerates rates. If the first $r$ moments of summands match with Gaussian, then the $\chi_2$–distance decays as $O(n^{-(r-1)/2})$, reflecting spectral gap phenomena in Hermite polynomial expansions (Delplancke et al., 2017).

---

Berry–Esseen rates provide microscopic control of CLT convergence speed across probabilistic regimes, from independent to highly dependent, finite-dimensional to operator-valued, commutative and non-commutative. The optimality of $O(n^{-1/2})$ persists under finite third or fourth moments in numerous settings, while functional, dependency, and non-commutative structures often introduce log-factors or yield lower polynomial rates. Matching leading moments or adopting symmetrizations (as in smooth-test Berry–Esseen bounds) can further improve constants and uncover the sharp limits of normal approximation phenomena.