I will first recall what these inequalities are in the context of dynamical systems, focusing in particular on the Gaussian bound, what they imply (mixing, multiple mixing, etc.), as well as various applications to observables that are not ergodic sums (for example, the distance between the empirical measure and the "true" invariant measure).

In a second part, I will explain why a random process -- or, equivalently, a shift-invariant measure on a product space -- that is a finitary coding of an i.i.d. process (i.e., of a product measure) satisfies a Gaussian concentration bound whenever the coding radius has finite second moment.

This is recent joint work with S. Gallo and D. Takahashi, which actually extends to random fields and has implications for classical models of statistical mechanics, such as the Ising model or the Potts model.