High-dimensional central limit theorems have been intensively studied with most focus being on the case where the data is sub-Gaussian or sub-exponential. However, heavier tails are omnipresent in practice. In this article, we study the critical growth rates of dimension $d$ below which Gaussian approximations are asymptotically valid but beyond which they are not. We are particularly interested in how these thresholds depend on the number of moments $m$ that the observations possess. For every $m\in(2,\infty)$, we construct i.i.d. random vectors $\textbf{X}_1,...,\textbf{X}_n$ in $\mathbb{R}^d$, the entries of which are independent and have a common distribution (independent of $n$ and $d$) with finite $m$th absolute moment, and such that the following holds: if there exists an $\varepsilon\in(0,\infty)$ such that $d/n^{m/2-1+\varepsilon}\not\to 0$, then the Gaussian approximation error (GAE) satisfies $$ \limsup_{n\to\infty}\sup_{t\in\mathbb{R}}\left[\mathbb{P}\left(\max_{1\leq j\leq d}\frac{1}{\sqrt{n}}\sum_{i=1}^n\textbf{X}_{ij}\leq t\right)-\mathbb{P}\left(\max_{1\leq j\leq d}\textbf{Z}_j\leq t\right)\right]=1,$$ where $\textbf{Z} \sim \mathsf{N}_d(\textbf{0}_d,\mathbf{I}_d)$. On the other hand, a result in Chernozhukov et al. (2023a) implies that the left-hand side above is zero if just $d/n^{m/2-1-\varepsilon}\to 0$ for some $\varepsilon\in(0,\infty)$. In this sense, there is a moment-dependent phase transition at the threshold $d=n^{m/2-1}$ above which the limiting GAE jumps from zero to one.

翻译：高维中心极限定理已被广泛研究，但大多数工作集中于数据服从次高斯或次指数分布的情形。然而，实践中重尾现象普遍存在。本文研究了高斯逼近渐近有效但超出该范围则失效的临界维度$d$增长率，重点关注这些阈值如何依赖于观测值所拥有的矩阶数$m$。对于每个$m\in(2,\infty)$，我们构造了$\mathbb{R}^d$中的独立同分布随机向量$\textbf{X}_1,...,\textbf{X}_n$，其各分量独立且服从同一分布（与$n$和$d$无关），该分布具有有限的$m$阶绝对矩，并满足：若存在$\varepsilon\in(0,\infty)$使得$d/n^{m/2-1+\varepsilon}\not\to 0$，则高斯逼近误差（GAE）满足$$ \limsup_{n\to\infty}\sup_{t\in\mathbb{R}}\left[\mathbb{P}\left(\max_{1\leq j\leq d}\frac{1}{\sqrt{n}}\sum_{i=1}^n\textbf{X}_{ij}\leq t\right)-\mathbb{P}\left(\max_{1\leq j\leq d}\textbf{Z}_j\leq t\right)\right]=1,$$ 其中$\textbf{Z} \sim \mathsf{N}_d(\textbf{0}_d,\mathbf{I}_d)$。另一方面，Chernozhukov等人(2023a)的结果表明，若仅存在$\varepsilon\in(0,\infty)$使得$d/n^{m/2-1-\varepsilon}\to 0$，则上式左侧为零。在此意义上，阈值$d=n^{m/2-1}$处存在一个矩依赖的相变：当维度超过该阈值时，极限GAE从零跳变为一。