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 $t,s>0$? 



How to construct a Poisson-like process with dependent increments?
Take a two dimensional poisson process and and make N(t) the number of events in some shape where the increments in the shape overlap, and the shape at time time has area t.
You can find a discussion of 2 dimensional poisson processes in Wikipedia, <https://en.wikipedia.org/wiki/Poisson_point_process>. A particular implementation of what I am suggesting is to take A(t) to be a triangle with vertices (0,0), (t,0), (t, 2), which has area t. The 'poisson process' is the number of events is this region, and it is by definition poisson with pararameter t at any fixed time. If you look at the joint distribution between N(1) and N(2), every important piece can me made out of 3 pieces. I = a tiringle with vertices (0,0). (t,0) , (1,1), II = triangle with vertices (0,0),(1,1), (1,2) and III = quadrilateral with vertices (1,0), (1,1), (2,2) , (2,0). These areas are disjoint and therefore the number of points in them are independent with poisson distribution whose parameter is equal to the area of the region. Let $X\_1, X\_2, X\_3$ be the number of events in each region. Then N(1) = # events in A(1) = $X\_1 + X\_2$, N(2) = $X\_1 + X\_3$, N(2) - N(1) = $X\_3 - X\_2$ which is not independent of N(1), as you can check by computing the covariance, or otherwise.


