Stationary increments are not a fixed level
A process N(t) counts arrivals during (0,t]. Explain why stationary increments do not imply that N(t) has a distribution free of t.
Worked answer (click to show)
Stationary increments fix the law of N(t+h)-N(t) for a given length h. They do not force the levels N(t) to share one distribution. A homogeneous Poisson process of rate \lambda has stationary increments, and N(t)\sim\mathrm{Poisson}(\lambda t) still depends on t.