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Without further assumptions you can't do better than the union bound (which should be $n e^{-\epsilon^2}$ as you've written things). If $X_i$ are identically distributed and the events $(|X_i| > \eps …

Let's try again. The theorem of Otto and Villani implies that every distribution $\nu$ which is log-concave in the sense you define satisfies a transportation-cost inequality. There are many distrib …

A different symmetrization-based proof is given in this review article by Schechtman (pp. 7-8); see the previous page for references.

(A) Nathael Gozlan proved that a distribution $\mu$ satisfies a $T_2$ inequality if and only if all finite tensor products of $\mu$ satisfy a subgaussian concentration property. (B) and (B') For fini …

It depends on exactly what you mean by "of the form ($*$)". As Davide points out (and as you certainly know if you've been reading Boucheron, Lugosi, and Massart), for centered subgaussian random var …