Concepts: time homogenous and independent increments Can someone give me an illustrative example for a time homogenous process without independent increments and for a process that is not time homogenous, but has independent increments?

First case: $X_n=(-1)^nU$ with $U$ symmetric, not deterministic. In continuous time, $X_t=\cos(t+2\pi V)$ with $V$ uniform on $(0,1)$.

Second case: $X_n=x_n$ with the sequence $(x_n)$ deterministic and not constant. In continuous time, $X_t=x_t$ with the function $x:t\mapsto x_t$ deterministic and not constant.