When is the Student $t$-statistic asymptotically standard normal?

Let $X, X_i, i \in \mathbb{N}$, be independent, identically distributed random variables. It is shown that the Student $t$-statistic based upon the sample ${X_i}_{i=1}^n$ is asymptotically $N(0, 1)$ if and only if $X$ is in the domain of attraction of the normal law. It is also shown that, for any $X$, if the self-normalized sums $U_n := \sum_{i=1}^n X_i/(\sum_{i=1}^n X_i^2)^{1/2}, n \in \mathbb{N}$, are stochastically bounded then they are uniformly subgaussian that is, $\sup_n \mathbb{E} \exp (\lambda U_n^2) < \infty$ for some $\lambda > 0$.