I failed to construct a process $N(t)$, $t\geq 0$, satisfying all the following three conditions for some positive $\lambda$:

1. $N(0)=0$,
2. for every $t$, $N(t)$ has Poisson distribution with parameter $\lambda t$
3. there are two number $s,t>0$ such that $N(t)$ and $N(t+s)-N(t)$ are not independent.

Does someone know how to construct such an example?

I failed to construct a process $N(t)$, $t\geq 0$, satisfying all the following three conditions for some positive $\lambda$:

1. $N(0)=0$,
2. for every $t$, $N(t)$ has Poisson distribution with parameter $\lambda t$
3. there are two number $s,t>0$ such that $N(t)$ and $N(t+s)-N(t)$ are not independent.

Does someone know how to construct such an example?

Update

How to construct such a process with surely non-negative increments, i.e. P(N(t+s)-N(t)<0)=0, for every $t,s>0$?