We consider random walks on lattices with finite memory and a finite number of possible steps. Using a local limit theorem, we generalize Polya's theorem to such walks, describe how to compute tail probabilities when the number of steps is large, and obtain asymptotic estimates for the average number of points visited.
"Correlated Random Walks." Ann. Probab. 12 (1) 274 - 278, February, 1984. https://doi.org/10.1214/aop/1176993392