Central Limit Theorems for Non-Invertible Measure Preserving Maps
  We establish a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic, using the Perron-Frobenius operator. We apply the result to asymptotically periodic transformations and give an extensive specific example of asymptotically periodic transformations by using the tent map. 
Michael C. Mackey and Marta Tyran-Kaminska