Oseledec's Multiplicative Ergodic Theorem is used to give recurrence and transience criteria for random walk in a random environment on the integers. These criteria generalize those given by Solomon in the nearest-neighbor case. The methodology for random environments is then applied to Markov chains with periodic transition functions to obtain recurrence and transience criteria for these processes as well.
"Recurrence and Transience Criteria for Random Walk in a Random Environment." Ann. Probab. 12 (2) 529 - 560, May, 1984. https://doi.org/10.1214/aop/1176993304