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, 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 $T$. 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?