  Understanding the statistical properties of the aperiodic planar Lorentz gas
stands as a grand challenge in the theory of dynamical systems. Here we study a
greatly simplified but related model, proposed by Arvind Ayyer and popularized
by Joel Lebowitz, in which a scatterer configuration on the torus is randomly
updated between collisions. Taking advantage of recent progress in the theory
of time-dependent billiards on the one hand and in probability theory on the
other, we prove a vector-valued almost sure invariance principle for the model.
Notably, the configuration sequence can be weakly dependent and non-stationary.
We provide an expression for the covariance matrix, which in the non-stationary
case differs from the traditional one. We also obtain a new invariance
principle for Sinai billiards (the case of fixed scatterers) with
time-dependent observables, and improve the accuracy and generality of existing
results.
