  We study the asymptotic behaviour of a random walk whose evolution is
dependent on the state of an itself dynamically evolving environment. In
particular, we extend our previous results in [Bethuelsen and V\"ollering,
2016] and prove a strong law of large numbers and large deviation estimates
assuming that the dynamic environment is "path-cone"-mixing. Under a mild
assumption on the decay rate of this mixing property we further obtain a
functional central limit theorem under the annealed law. Our method of proofs
rest on the study of the so-called local environment process and general
results for $\phi$-mixing stochastic processes.
