Let $\psi_\alpha(x) := \exp(x^\alpha)-1$. It is well-known that for $\alpha\geq 1$ that
$$\lVert X\rVert_{\psi_\alpha} = \inf\{k>0\mid \mathbb{E}[\psi_\alpha(|X|/k)] \leq 1\}$$
defines an Orlicz norm on the space of random variables. The norm for $\alpha = 2$ is known as the sub-Gaussian norm, and $\alpha = 1$ is known as the sub-Exponential norm. It is also known that $\lVert XY\rVert_{\psi_1} \leq \lVert X\rVert_{\psi_2}\lVert Y\rVert_{\psi_2}$ (perhaps up to an absolute constant), e.g. one may control the sub-Exponential norm of a product via the sub-Gaussian norm of the product-ands. For random vectors, one defines
$$\lVert \vec X\rVert_{\psi_a} = \sup_{\theta : \lVert \theta\rVert_{2} = 1} \lVert \langle \vec X, \theta\rangle\rVert_{\psi_a}.$$
Let $n\in\mathbb{N}$. Let $\mathcal{D}$ be a distribution of finite sub-Gaussian norm on $\mathbb{R}[x]/(x^n-1)\cong \mathbb{R}^n$, where the isomorphism is the mapping of coefficients of polynomials to vectors in $\mathbb{R}^n$. I am curious about what one can say about $\lVert f(x)g(x)\rVert_{\psi_a}$.
By noticing that any particular coordinate of $f(x)g(x)$ is the sum of $n$ products of independent sub-Gaussian random variables, it is straightforward to get the estimate
$$\lVert f(x)g(x)\rVert_{\psi_1}\leq n\lVert f(x)\rVert_{\psi_2}\lVert g(x)\rVert_{\psi_2}.$$
I am curious if you can do better somehow. For example, if $\mathcal{D}$ is not solely sub-Gaussian, but is the product of $n$ 1D independent sub-Gaussian random variables, then any particular coordinate of $f(x)g(x)$ is the sum of $n$ independent random variables. One would expect that for large $n$ this will (by the CLT) approach a Gaussian, and therefore would have Gaussian-like tails.
Can this be formalized/made quantitative at all, via some bound on $\lVert f(x)g(x)\rVert_{\psi_2}$? Or (in general) via some stronger Orlicz-norm bound on $f(x)g(x)$ than the above $\psi_1$ bound?
