The statistical properties of the Lorentz gas with periodically positioned obstacles are well understood. The random case, obtained after each of the obstacles undergoes a small i.i.d. displacement, stands as a challenge. The latter can be studied in terms of a random sequence of hyperbolic symplectic (billiard) maps, which however is not i.i.d. due to recollisions. In fact, even the i.i.d. sequence (no recollisions) is poorly understood.Motivated by the above, we study an i.i.d. sequence of toral automorphisms in two dimensions. We will argue that the time-$N$ average of any observable has Gaussian fluctuations of order $\sqrt{N}$ for almost every sequence of maps, and that the variance is independent of the sequence. Joint work with Arvind Ayyer (Rutgers) and Carlangelo Liverani (Rome 1).