Abstract. 
Understanding the statistical properties of the aperiodic
planar Lorentz gas stands as a grand challenge in the theory of
dynamical systems. We study a greatly simplified but related model,
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
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.