Classical results concerning concentration of Gaussian random variables due to Cirelson, Ibragimov and Sudakov say that if $V_1,\cdots,V_n$ are jointly Gaussian with variance bounded by $1$, then (constant can be improved but not much of my concern) \begin{equation} \mathbb{P}(|M-\mathbb{E}[M]|>t)\leq 2\exp(-t^2/2) \end{equation} where $M=\max_{1\leq i\leq n}V_i$. Similar dimension-free concentration results hold for suprema of Gaussian process indexed by general $T$. This follows by the general log-Sobolev inequality for Gaussian measures. For general processes, the approach I'm aware of involves some kind of maximal inequality(c.f. van de Vaart and Wellner(1996)), where in the concentration bound an entropy number measuring the size of $T$ is necessary.
My question is that, is the dimension free concentration phenomenon unique to Gaussian processes, or even when $|T|<\infty$, do we have analogous result to the above display when $V_i$'s are non-Gaussian? In particular, I'm curious if there is any analogous result of this type with (possibly) $$\mathbb{P}(|M-\mathbb{E}[M]|>t)\leq C t^{-\alpha} \wedge 1$$ for some $\alpha>0$ that may be determined by moment conditions of $V_i$'s?
