A strictly stationary, N-tuplewise independent counterexample to the   central limit theorem
  For an arbitrary integer N that is at least 2, this paper gives a construction of a strictly stationary, N-tuplewise independent sequence of (non-degenerate) bounded random variables such that the Central Limit Theorem fails to hold. The sequence is in part an adaptation of a non-stationary example with similar properties constructed by one of the authors (ARP) in a paper published in 1998. 
Richard C. Bradley and Alexander R. Pruss