Necessary and sufficient conditions for the self-normalized central limit theorem for independent triangular arrays
Li et al.
For rowwise independent triangular arrays, the self-normalized central limit theorem is shown to force negligibility of the largest term, which yields necessary and sufficient conditions and settles a conjecture of Shao.
Source published · In the Daily Brief of 9 November 2026
Filed under Probability
Write \(S_n=\sum_{i=1}^n X_{n,i}\) and \(V_n^2=\sum_{i=1}^n X_{n,i}^2\) for a rowwise independent array. The authors prove that \(S_n/V_n\to N(0,1)\) in distribution implies \(\max_{i\le n}|X_{n,i}|/V_n\to 0\) in probability and \(V_n^2/b_n^2\to 1\) in probability for any choice of medians \(b_n^2\) of \(V_n^2\).
Combined with results of Shao from 2018, this gives 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 of its sum of squares, then uses the standard normal ratio to remove every non-Gaussian component.