When is the Student t-statistic asymptotically standard normal?

Let X, Xᵢ, i ∈ N, be independent, identically distributed random variables. It is shown that the Student t-statistic based upon the sample Xᵢᵢ₌₁ⁿ 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 Uₙ := ∑ᵢ₌₁ⁿ Xᵢ/(∑ᵢ₌₁ⁿ Xᵢ²)1/2, n ∈ N, are stochastically bounded then they are uniformly subgaussian that is, ₙ E exp (λ Uₙ²) < ∞ for some λ > 0.