Esseen's Anti-Concentration Bound in Tensors

• Esseen's anti-concentration bound is a precise quantitative measure of the Kolmogorov distance between a normalized tensor sum and a Gaussian distribution.
• The framework leverages intrinsic parameters such as oscillation, partial-sum seminorms, and correlation measures to extend classical limit theorems to high-dimensional and degenerate settings.
• The approach unifies results from Esseen, Bolthausen, and Barbour–Chen by using exchangeable-pair coupling and Stein’s method to provide robust error estimates.

The anti-concentration bound of Esseen describes a quantitative, non-asymptotic estimate for the Kolmogorov distance between a normalized statistic and the Gaussian, highlighting the extent to which a linear combination of symmetric, exchangeable random variables (or tensors) avoids being too concentrated around any single value. The work of Dodos–Tyros (Dodos et al., 2022) establishes sharp anti-concentration (Berry–Esseen-type) bounds for sums of the form $$S = \langle \theta, X \rangle = \sum_{i\in[n]^d} \theta_i X_i$$, where $X$ is a random tensor with strong symmetry, and provides explicit error terms in terms of intrinsic parameters. This framework recovers and extends classical results—such as Esseen’s and Bolthausen’s theorems—to arbitrary tensor order $d$, encompassing high-dimensional and degenerate regimes and elucidating the transition between classical independence and combinatorial dependence structures.

1. Framework and Statement of the Anti-Concentration Bound

Let $n,d \in \mathbb{N}$; $X = (X_i : i \in [n]^d)$ a real-valued random tensor; and $\theta = (\theta_i : i\in[n]^d)$ a deterministic tensor (vector of coefficients). The main object is the sum

$$S = \langle \theta, X \rangle = \sum_{i\in[n]^d} \theta_i X_i$$

Let $\sigma^2 = \operatorname{Var}(S)$ and suppose a nondegeneracy condition on a parameter $\delta_1$:

$$\delta_1 > \max \left\{ \operatorname{osc}(X)^{\alpha},\; B^\alpha,\; \left(\frac{\kappa}{n}\right)^\alpha \right\}$$

for some $\alpha \in (0,1)$, with $\kappa = 20d^3 18^d (2d)!$ and $B$, $\operatorname{osc}(X)$ as defined below.

The anti-concentration (Kolmogorov) bound [Theorem 1.4; (Dodos et al., 2022)] is:

$d_K(S, N(0, \sigma^2)) \leq E_1 + E_2 + E_3$

with $E_1$, $E_2$, $E_3$ given explicitly by: $$\begin{align*} E_1 &= 5\,\operatorname{osc}(X)^{1-\alpha} + 5\,|\delta_0|^{1-\alpha} + \left| \frac{\delta_0(\|\theta\|_0^2 - 1)}{d^2 \delta_1} \right| + \frac{6\kappa}{n^{1-\alpha} + 4\|\theta\|_0^2 / n} \ E_2 &= \frac{2^{36} \mathbb{E}[|X_{(1,\ldots,d)}|^3]}{ \delta_1^{3/2} \cdot \sum_{j=1}^n \left| \sum_{i(1)=j} \theta_i \right|^3 } \ E_3 &= \frac{3\kappa}{ d \sqrt{\delta_1} \cdot \sum_{s=2}^d \binom{d}{s} \sqrt{s!} \sqrt{ \Sigma_s + 16 d^2 2^d / n }\, \|\theta\|_s } \end{align*}$$

2. Parameter Definitions

The error terms depend on several intrinsic parameters and seminorms:

• $s$-Partial Sum Seminorm: For $0\leq s \leq d$,

$$\|\theta\|_s = \left( \sum_{j\in[n]^s} \left( \sum_{j\sqsubseteq i\in[n]^d} \theta_i \right)^2 \right)^{1/2}$$

Special cases: $\|\theta\|_0 = |\sum_{i}\theta_i|$ (total sum), $\|\theta\|_d = \|\theta\|_2$ (Euclidean norm).

• Exchangeability/Correlation Parameters $\delta_t$: For $0 \leq t \leq d$,

$$\delta_s = \mathbb{E}\left[ X_{(1,\ldots,d)} X_{(1,\ldots,s,\,d+1,\ldots,2d-s)} \right]$$

Notably, $$\delta_0 = \mathbb{E}[X_{(1,\ldots,d)} X_{(d+1,\ldots,2d)}]$$.

• Finite-population Hoeffding analogues $\Sigma_s$: For $0\leq s\leq d$,

$$\Sigma_s = \sum_{t=0}^s (-1)^{s-t} \binom{s}{t} \delta_t$$

• Oscillation: The L$_1$ deviation of coordinate block averages,

$$\operatorname{osc}(X) = \left\| n^{-1}\sum_{j=1}^n \left[ n^{-(d-1)}\sum_{i(1)=j} X_i \right]^2 - \delta_1 \right\|_{L_1}$$

• Global Mean-Deviation: $B = \| n^{-d} \sum_i X_i \|_{L_2}^2$
• Explicit Constant: $\kappa(d) = 20 d^3 18^d (2d)!$

3. Structural Hypotheses and Nondegeneracy

The bound requires the following properties for $X$ and $\theta$:

• (A1) Moment and variance control: $\mathbb{E}[X_i]=0$, $\mathbb{E}[X_i^2]\leq 1$, $\mathbb{E}|X_i|^3<\infty$
• (A2) Symmetry, exchangeability, diagonal-free:
  • $X_{i_1\ldots i_d}$ invariant under coordinate permutations
  • Distribution of $X$ invariant under any permutation of $[n]$
  • $X_{i}=0$ if $i$ has repeated coordinates
• (A3) Identical symmetry and diagonal-free assumptions for $\theta$
• Nondegeneracy: $\delta_1$ must be bounded below as specified above to avoid division by nearly zero denominators.

These structural conditions generalize beyond the i.i.d. setup to highly dependent, symmetric arrays where standard independence-based CLTs do not apply.

4. Connections to Classical Esseen, Bolthausen, and Barbour–Chen Bounds

The Dodos–Tyros bound generalizes several pivotal prior results:

• i.i.d. (Esseen/Berry–Esseen):

$$d_K\left( \frac{\sum X_i}{\sqrt{n}}, N(0,1) \right) = O \left( \frac{ \mathbb{E}|X|^3 }{ \sigma^3 n^{1/2} } \right)$$

for third-moment finite, independent entries.

• Bolthausen’s combinatorial CLT: For order-1 permutation statistics, e.g. sums $\sum \xi(i, \pi(i))$, the optimal rate is

$$d_K(\text{statistic}, N(0,1)) = O \left( n^{-1}\sum |\xi(i,j)|^3 \right)$$

• Barbour–Chen: For two-dimensional permutation U-statistics, the bound is

$$d_K(W, N(0,1)) = O(n^{-1} \|\xi_1\|_3^3) + O(\operatorname{Var}(\xi_2)/n)$$

For $d=1,2$, the Dodos–Tyros $E_2$ term matches the “linear” Berry–Esseen rates, while $E_3$ captures the degenerate variance–ratio perturbation, merging these regimes in a unified framework. For $d>2$, the bound accommodates further tensor structure.

5. Combinatorial Central Limit Theorem for High-Dimensional Tensors

The key methodological advance is a combinatorial CLT tailored to random tensors and permutation statistics [Theorem 2.2]:

Let $\xi_s: [n]^s \times [n]^s \to \mathbb{R}$ be Hoeffding-type symmetric, zero-average kernels with $\beta_s = \sum_{i,p\in[n]^s} \xi_s(i,p)^2$ and $\beta_1 = n-1$. The statistic

$$W = \sum_{s=1}^d \sum_{i \in [n]^s_{\mathrm{Inj}}} \xi_s(i, \pi(i))$$

where $\pi \sim \mathrm{Uniform}(\mathbb{S}_n)$, obeys

$$d_K(W,N(0,1)) \leq \frac{2^{18} C_1}{n} \sum_{i,j} |\xi_1(i,j)|^3 + C_d \sum_{s=2}^d \sqrt{ \frac{\beta_s}{n^s} }$$

with $C_1 \approx 451$ (Bolthausen’s constant), $C_d = 5d^2 e^d (2d)!$.

The proof constructs an exchangeable-pair coupling $(\pi_1, \pi_2)$ via random transpositions and exploits Stein’s method—ensuring the required linearity and variance control—supported by a generalized Hoeffding multi-index variance decomposition and direct moment bounds. The approach extends Barbour–Chen’s Stein concentration-inequality methods to high-rank tensors.

6. Optimal Regimes and Theoretical Implications

Sharpness and optimality of the anti-concentration bound depend on intricate relationships between independence, degeneracy, and symmetry:

• Oscillation-dominated regime: If $\operatorname{osc}(X)$ is large (weak dissociation), $E_1$ dominates and cannot be improved beyond $O(\operatorname{osc}(X))$.
• Nearly-linear regime: If $X_i$ are almost independent and all $\delta_0, \delta_s \approx 0$ (for $s \geq 2$), $E_2$ matches Bolthausen’s $O(n^{-1} \sum |\theta_i|^3)$ term.
• Partially degenerate regime: If degeneracy at some $s \geq 2$ is present but small, $E_3$ yields an error smaller than $O(n^{-1/2})$, as in U-statistics of small effective rank.
• High-dimensional/mixed regime: For large $d$, regimes interpolate smoothly between independence and fully degenerate (Hoeffding) structures.

For i.i.d. entries, the bound recovers the classical $O(n^{-1/2})$ Berry–Esseen rates when $\|\theta\|_1$ is small, and Bolthausen’s $O(n^{-1})$ rate when $\sum \theta_i = 0$. Fully degenerate U-statistics ($\delta_1 = \cdots = \delta_{d-1} = 0$) achieve even faster rates in small effective-rank settings.

7. Significance and Extensions

The anti-concentration bound of Esseen for random tensors unifies and extends statistical normal approximation in highly symmetric, exchangeable, and high-dimensional settings. The explicit dependence on the oscillation, correlation, mean-deviation, and partial-sum seminorms provides practically computable error estimates that are minimax optimal in several key regimes. This anti-concentration framework is instrumental in analyzing linear and nonlinear permutation statistics, and provides a rigorous foundation for statistical inference in combinatorial and high-order data analytic scenarios (Dodos et al., 2022).