Self-normalized central limit theorem with a second-moment denominator
= Self-normalized central limit theorem with a second-moment denominator
For independent identically distributed $Z_i$ with mean zero and variance one,
$$
\frac{n^{-1/2}\sum_{i=1}^nZ_i}{n^{-1}\sum_{i=1}^nZ_i^2}
\xrightarrow dN(0,1).
$$
The numerator obeys the central limit theorem, the denominator converges almost surely to one by the strong law, and Slutsky's theorem combines them.