A real random variable $X$ is said to be *subgaussian* if there exists an $a > 0$ such that $\mathbb{E}[e^{\lambda X}] < e^{a^2 \lambda^2}$ for all $\lambda \in \mathbb{R}$. The space of such random variables admits a Banach space structure, with an Orlicz norm given by $$\| X \|_{\psi_2} = \inf\left\{ t > 0 : \mathbb{E}\left[ \psi_2\left( \frac{|X|}{t} \right) \right] \leq 1 \right\},$$ with $\psi_2(x) = e^{x^2} - 1$.

The main question here is a simple one to ask, but I've been unable to find an answer. Suppose I have a sequence of i.i.d. subgaussian random variables. Is it known whether or not the normalized sums $$S_n = \frac{1}{\sqrt{n}} \sum_{i=1}^n X_i,$$ converge in this norm to a normally distributed random variable? Moreover, if they do not do so in general, is a sufficient condition known?