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

Xiang Li, Song-Hao Liu, Qi-Man Shao +2

For a rowwise independent triangular array $\{X_{n,i}:1\le i\le n\}$, put $S_n={\sum }_{i=1}^n{X}_{n,i}$ and $V_n^2={\sum }_{i=1}^n{X}_{n,i}^2$. We prove that $S_n/V_n\xrightarrow{\mathrm d}…