relation between stationary and independent increment

Question

I am a little confused about the relation between stationary and independent increment. I think if a process does not have independent increment, it does have stationary increment either. Is that right.

Answer 1

For a general stochastic process, fractional brownian motion (with $H\neq 1/2$) would be a counterexample. It doesn't have independent increments, but it does have stationary increments.

For a counting process, you can get a counterexample by letting $N(t) : [0,\infty) \to \mathbb{N}$ denote a poisson process and then define $f(t) : [0, \infty) \to \mathbb{N}$ by $$f(t) = \lfloor t\rfloor N(1) + N(\{t\})$$ It's easy to see this doesn't have independent increments (e.g. $f(2)-f(1) = f(1) - f(0)$), but it has stationary increments.