We construct examples of a random walk with pairwise-independent steps which is almost surely bounded, and for any m and k a random walk with k-wise independent steps which has no stationary distribution modulo m.
Itai Benjamini. Gady Kozma. Dan Romik. "Random walks with k-wise independent increments." Electron. Commun. Probab. 11 100 - 107, 2006. https://doi.org/10.1214/ECP.v11-1201