Quantitative Central Limit Theorem (QCLT)

• QCLT is a framework that defines explicit convergence rates for normalized sums of random variables to a Gaussian limit using analytic and PDE-based methods.
• It refines classical CLT by quantifying the influence of sample size, smoothness of test functions, and moment bounds to achieve sharp error estimates.
• The theorem’s techniques are applicable to multidimensional settings, offering practical insights for statistical physics, random geometry, and advanced probability analysis.

The Quantitative Central Limit Theorem (QCLT) provides explicit convergence rates for normalized sums of random variables (or more generally, statistics) to their Gaussian limit. It refines classical CLT statements by specifying dependence on parameters such as sample size, smoothness of test functions, moment bounds, and structural properties of the underlying model. Modern QCLT theory leverages analytic, probabilistic, and PDE-inspired techniques to deliver sharp rates and deep connections across probability, statistical physics, random geometry, and analysis.

1. Formulation of Quantitative CLT Results

Consider iid mean-zero $\mathbb{R}^d$-valued random vectors $X_1, X_2, \dots$ with covariance $\Sigma>0$ and finite moment $E[|X_1|^{2+\gamma}]<\infty$ for some $\gamma \in (0,1]$. The normalized sum $S_n = \sum_{i=1}^n X_i$, $S_n/\sqrt{n}$, approximates the Gaussian $\xi \sim N(0,\Sigma)$. For a test function $f \in C^4(\mathbb{R}^d)$ vanishing at infinity, the QCLT bound reads: $$\left| E\left[f\left(\frac{S_n}{\sqrt{n}}\right)\right] - E[f(\xi)] \right| \leq C \,E[|X_1|^{2+\gamma}]\, n^{-\gamma/2}$$ where $C$ depends on $d$, $\|f\|_{C^4}$, and $\lambda_{\max}(\Sigma)$ (Addario-Berry et al., 2022). For $\gamma=1$, this coincides with a Berry–Esseen-type bound for smooth test functions in $\mathbb{R}^d$.

2. Analytical Techniques: Doubling-of-Variables and PDE Approach

The proof strategy via "doubling of variables" capitalizes on embedding the probabilistic problem into a discrete analogue of the heat equation:

• The average $$u_n(x,t) = E[f(x + S_{\lfloor nt \rfloor}/\sqrt{n})]$$ satisfies a finite-difference scheme,
• This discrete scheme aligns with the continuous PDE $$\partial_t u = \tfrac{1}{2}\mathrm{tr}(\Sigma D_x^2 u)$$,
• A penalized maximization over the 'doubled' test function $\Phi_n(x,t,s)$ ensures that the main contribution arises from neighborhoods $|t-s| = O(n^{-1/2})$,
• Application of Taylor expansions and moment bounds yields the desired $n^{-\gamma/2}$ scaling.

This approach is dimension-agnostic and bypasses characteristic function calculations, making it suitable for multidimensional and structured random vector settings (Addario-Berry et al., 2022).

3. Comparison to Classical Berry–Esseen Theorems

Traditionally, the multivariate Berry–Esseen theorem states: $$\sup_{A \text{ convex}} |P(S_n/\sqrt{n} \in A) - P(\xi \in A)| \leq K(d) \,E[|X_1|^3]\, n^{-1/2}$$ where $K(d)$ increases polynomially with $d$ (Addario-Berry et al., 2022). The QCLT in the PDE/doubling formalism matches the $n^{-1/2}$ rate for smooth $f$, but does not directly capture non-smooth metrics such as the Kolmogorov distance or indicators of convex sets. However, smoothing or approximation arguments can bridge this gap at the expense of small additional errors. Unlike Berry–Esseen's reliance on characteristic functions, the PDE-based technique is fully analytic and constructive.

4. Generalizations: Nonlinear, Dependent, and Non-Smooth Statistics

• The doubling-of-variables method, as currently formulated, requires independence and identical distribution.
• Extensions to non-identical or weakly dependent variables may be feasible by tracking how the generator discrepancies enter the error terms.
• For self-normalized sums and other nonlinear statistics, a suitable nonlinear PDE should replace the heat equation in the comparison analysis.
• Handling discontinuous test functions (e.g., indicator functions) necessitates additional smoothing, and the constants may deteriorate or incur extra error terms (Addario-Berry et al., 2022).

5. Structure of Error Bounds and Dimension Dependence

The rate exponent $-\gamma/2$ interpolates smoothly between the classical CLT ($\gamma \to 0$) with no explicit rate and Berry–Esseen ($\gamma = 1$) for finite third moments:

• Smoothness of the test function ($C^4$) is essential for optimal rates and for leveraging the Taylor expansion framework.
• The dimension dependence of constants remains an open problem; the bounds are explicit but sharp analytic estimates on $C(d,\Sigma)$ have not been resolved (Addario-Berry et al., 2022).
• The role of the maximal covariance eigenvalue $\lambda_{\max}(\Sigma)$ enters directly in the error bound scaling.

6. Significance and Outlook

The QCLT as formulated through the doubling-of-variables and PDE method achieves:

• A uniform, explicit family of rates $n^{-\gamma/2}$ under minimal finite-moment assumptions,
• Applicability to $\mathbb{R}^d$-valued random vectors without further structure beyond covariance non-degeneracy,
• An analytic proof pathway importing techniques from PDE uniqueness theory into probability,
• A platform for future investigation of sharper constants, non-smooth statistics (via approximation), and possible non-linear and weakly dependent regime generalizations.

No improvement in the exponent over classical Berry–Esseen results is claimed, but the QCLT provides enhanced analytic clarity and universality in settings suitable for PDE and monotone-scheme methods in probability.

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References:

"The central limit theorem via doubling of variables," (Addario-Berry et al., 2022)