---
title: Esseen's Anti-Concentration Bound in Tensors
url: https://www.emergentmind.com/topics/anti-concentration-bound-of-esseen
type: topic
---

# Esseen's Anti-Concentration Bound in Tensors

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 [2205.00708] 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; 2205.00708] 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 [2205.00708].

Source: https://www.emergentmind.com/topics/anti-concentration-bound-of-esseen