Counting process which is not a Poisson process Please construct a counting process N, whose r.v. N(t) are distributed as Poisson(λt) but the process N itself is not a Poisson process.  
This is an assignment in our Stochastic Process class. So I suppose this counting process N should meet all but one of a Poisson's process conditions. 1) N(0)=0 2) independent increments 3) At any given time t N(t) ~ Poiss( λt). So making the increments dependent should probably be the way to go. However, I've no idea how that could be done.
 A: Let $\{N_t, t\geq 0\}$ be a homogeneous Poisson process with intensity $\lambda$, it holds that $N_{t+s}-N_s\sim Pois(\lambda t)$. Thus, let $Z_t=N_{t+s}-N_{s}$ for any $t,s\geq 0$. Then clearly $\{Z_t, t\geq 0\}$ is a counting process, $Z_t\sim Pois(\lambda t)$, but most importantly - its increments are dependent, e.g. let $m>s\geq 0,\,t_2>t_1\geq 0$ and then $Z_{t_2}-Z_{t_1}=N_{t_2+s}-N_{t_1+s}$ and $Z_{t_1}-Z_{0}=N_{t_1+m}-N_{m}$, intervals $\left(t_1,t_2\right]$ and $(0,t_1]$ do not overlap, but $N_{t_2+s}-N_{t_1+s}$ and $N_{t_1+m}-N_{m}$ are dependent since $m>s$.
A: Let $N$ be a homogeneous Poisson Process on $[0,+\infty)$ and define $\tilde N$ by
\[ \tilde N(t) = N(2+t) - N(2) \text{ if } t\in [0,2),
\qquad \tilde N(t) = N(t) \text{ if } t\ge 2.
\]
Then $\tilde N$ satisfies $\tilde N(t) \sim Poisson(\lambda t)$ for every value of $t$ by studying separately $t<2$ and $t\ge 2$.
However there is a clear dependence because $\tilde N(1) = \tilde N(3)-\tilde N(2)$ almost surely, so for instance
\[P[\tilde N(1)=0, \tilde N(3) - \tilde N(2)=0]= P[\tilde N (1)=0]=e^{-\lambda}\]
but this quantity would be equal to $e^{-2\lambda}$ if the increments were independent.
A: Let $$N(t) \sim \text{Poisson}(\lambda t) \quad\forall t > 0$$
Let's examine $T_n$ which is the sum of the inter event times of a Poisson Process.
We know that $$T_n \sim \text{sum of n i.i.d } exp(\lambda)$$
$$\{T_n \leq t\} = \{N(t) \geq n\}$$ 
We also know that the sum of i.i.d exponentials follows the Erlang-n distribution.
Let $F_n(t) = \mathbb{P}\{T_n \leq t\} \sim \text{Erlang}(n,\lambda)$ and let $U \sim unif(0,1)$.
$$F_n^{-1}(U) \sim \text{Erlang}(n,\lambda)$$
Now if we define $$t_i = F_i^{-1}(U)$$ and examine the counting process on $t_i$.
$N(t) \sim \text{Poisson}(\lambda t)$ but all the times are dependent. If we know any one value we know the entire counting process and the independent increments property is lost!
