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?



The central limit theorem in the subgaussian Orlicz norm
$\newcommand\ep\varepsilon$No. If this were so, then (by Lemma 1 below) $S\_n$ would converge to a normally distributed random variable $Y$ in probability, which is [false](https://math.stackexchange.com/questions/2145140/clt-cannot-be-enhanced-to-convergence-in-probability) for any iid $X\_i$'s.




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> 
> **Lemma 1:** If $\|S\_n-Y\|\_{\psi\_2}\to0$ (as $n\to\infty$), then $S\_n\to Y$ in probability.
> 
> 
> 


*Proof:* Suppose that $\|S\_n-Y\|\_{\psi\_2}\to0$. Then
$$E\psi\_2(|S\_n-Y|/t\_n)\le1$$
for all natural $n$, where $t\_n:=\|S\_n-Y\|\_{\psi\_2}+1/n\to0$.
So, by Markov's inequality, for each real $\ep>0$,
$$P(|S\_n-Y|\ge\ep)\le\frac{E\psi\_2(|S\_n-Y|/t\_n)}{\psi\_2(\ep/t\_n)}
\le\frac1{\psi\_2(\ep/t\_n)}\to0.$$
So, $S\_n\to Y$ in probability. $\quad\Box$


