Limit laws and recurrence for the planar Lorentz process with infinite horizon

Bleher ( J. Stat. Phys. 66( 1): 315 - 373, 1992) observed the free flight vector of the planar, infinite horizon, periodic Lorentz process {S-n | n = 0, 1, 2,...} belongs to the non-standard domain of attraction of the Gaussian law - actually with the root n log n scaling. Our first aim is to establish his conjecture that, indeed, S-n/root n log n converges in distribution to the Gaussian law ( a Global Limit Theorem). Here the recent method of Balint and Gouezel (Commun. Math. Phys. 263: 461 - 512, 2006), helped us to essentially simplify the ideas of our earlier sketchy proof (Szasz, D., Varju, T. in Modern dynamical systems and applications, pp. 433 - 445, 2004). Moreover, we can also derive ( a) the local version of the Global Limit Theorem, (b) the recurrence of the planar, infinite horizon, periodic Lorentz process, and finally ( c) the ergodicity of its infinite invariant measure.