This is related to my previous question, but is hopefully more precise.
I would like to reason about tail-bounds for polynomial products of concentrated random variables in $R:=\mathbb{R}[x]/(x^n-1)$. Note that for $f(x), g(x)\in R$, we have that the $d$th coefficient of the product is
$$ \sum_{i\in[n]}f_ig_{d-i\bmod n}, $$ where $f_i, g_i$ are the coefficients of $f$ and $g$.
Briefly, by $\psi_\alpha$ random variable I mean a random variable of bounded Orlicz norm for the Young function $\psi_\alpha(x) = \exp(x^\alpha)-1$. $\psi_2$ corresponds to sub-Gaussian random variables. $\psi_1$ corresponds to sub-Exponential random variables.
It is well-known that if we start with the assumption that $f, g$ are $\psi_2$ random variables, we can naively prove a $\psi_1$ bound on the polynomial product. This has two main issues for me
The $\psi_1$ concentration is much weaker than one expects in practice, at least for $n$ large (so $f(x)g(x)$ will, under appropriate independence assumptions on the coefficeints of $f_i, g_i$ which are fine in my setting, will start approaching a Gaussian, e.g. $\psi_2$ concentration).
It is not composable. If we get a $\psi_1$ bound on the product $f(x)g(x)$, it does not help with analysis of $f(x)g(x)h(x)$ for $h(x)$ independent of $f, g$, and satisfying a $\psi_2$ bound.
It appears there are Orlicz norms that fix my issue #1 above. Namely, the content of Bernstein's inequality is that the sum of many $\psi_1$ random variables will have a (small deviation) $\psi_2$ tail, and (large deviation) $\psi_1$ tail, e.g. at least for some parameter range I can start with (assumed) $\psi_2$ tails, and get a $\psi_2$ bound for some region of the tail.
I am curious if this can additionally be made composable as well. There are many ways this question can be answered. One natural conjecture is the following though.
Let $L>0$. Define the $L$-Bernstein-Orlicz norm as the Orlicz norm associated with the function $\Psi_L(x) = \exp\left(\left[\frac{\sqrt{1+2Lz}-1}{L}\right]^2\right)-1$. Then, is there some function $h(x,y)$ such that for any independent random variables $f(x), g(x)$ on $R$ that $$\lVert f(x)g(x)\rVert_{\Psi_{h(L,L')}} \leq \lVert f(x)\rVert_{\Psi_L}\lVert g(x)\rVert_{\Psi_{L'}}$$
Essentially, if we start with full $\psi_2$ variables, we still get $\psi_2$ behavior near the mean of $f(x)g(x)$. If we instead start with $\Psi_L$ random variables (which are $\psi_2$ near their means), do we still get $\psi_2$ behavior near the mean of the random variables (perhaps for some concretely smaller notion of "near")? It feels like the answer should be yes, but I am not a probabilist, so am not sure if this is already known.
