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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 $t,s>0$?

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  • $\begingroup$ The second condition and normalization put necessary and sufficient conditions on $a_{k|\ell}:=P(N(t+s)=k|N(t)=\ell)$, $k,\ell\in\mathbb{N}$, in the form of countably many linear equations. They should yield more than one solution (the one that corresponds to independent increments) within the constrains $a_{k|\ell}\in [0,1]$. Yet, I am not able to prove it. $\endgroup$ Commented Dec 18, 2017 at 14:22

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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.

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  • $\begingroup$ If I well understand, you suggest to find Borel subsets $A_t\subset\mathbb{R}^2$, $t>0$ of special shapes where $A_t$ has area $t$. Then project two dimensional Poisson process to the one-dimensional. Can you give me, please, more hint? Where is the advantage of the use of two-dimensional process? Might I manage it with one-dimensional process and sets $A_t\subset \mathbb{R}$? Any idea I got leads to the case, when $N(t+s)-N(t)$ could be negative with positive probability. What to do when I add the condition of non-negativeness of the "increments"? $\endgroup$ Commented Dec 18, 2017 at 15:43
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    $\begingroup$ Thank you, your construction solved the original problem and give me an insight . The same is even possible to do with without the second dimension. But I should say I had in mind an example where the increments are non-negative, and if possible natural numbers. I should edit the question. $\endgroup$ Commented Dec 18, 2017 at 16:33

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