Petersburg Department of Steklov Institute of Mathematics

PREPRINT 05/2000

ON THE ALMOST SURE CENTRAL LIMIT THEOREM FOR A CLASS OF ${\Bbb Z}^d$-ACTIONS

This preprint was accepted May, 2000
Contact: M. Gordin

ABSTRACT:
As an extension of earlier papers on stationary sequences,
a concept of weak dependence for strictly stationary random
fields is introduced in terms of so-called homoclinic
transformations. Under assumptions made within the
framework of this concept a form of the almost sure central
limit theorem (ASCLT) is established for random fields
arising from a class of algebraic ${\Bbb Z}^d$-actions on compact
abelian groups. As an auxillary result, the central limit
theorem is proved via Ch. Stein's  method. The next stage
of the proof includes some estimates which are specific for
ASCLT.  Both steps are based on making use of homoclinic
transformations.