# The central limit theorem in the subgaussian Orlicz norm

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?

• Ah, apologies, I was unsure of the norm within the forum Mar 26, 2022 at 4:16

$$\newcommand\ep\varepsilon$$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 for any iid $$X_i$$'s.
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. $$\quad\Box$$