Necessary and sufficient conditions for the self-normalized central limit theorem for independent triangular arrays

Abstract

For a rowwise independent triangular array $\{X_{n,i}:1\le i\le n\}$, put $S_n=\sum_{i=1}^nX_{n,i}$ and $V_n^2=\sum_{i=1}^nX_{n,i}^2$. We prove that $S_n/V_n\xrightarrow{\mathrm d}{\mathrm N}(0,1)$ implies $\max_{i\le n}|X_{n,i}|/V_n\xrightarrow{\mathbb P}0$. The same assumption gives $V_n^2/b_n^2\xrightarrow{\mathbb P}1$ for any choice of medians $b_n^2$ of $V_n^2$. Combined with the results of Shao (2018), this yields necessary and sufficient conditions for the self-normalized central limit theorem and resolves his conjecture for independent sequences. The proof identifies the joint limits of the sum and its sum of squares, then uses the standard normal ratio to remove every non-Gaussian component.