Moment-dependent phase transitions in high-dimensional Gaussian approximations

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.