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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.

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Please provide some context regarding why you are considering such an object, what you've thought about and where you are stuck. – cardinal Apr 17 '12 at 11:18
suggestion: take a poisson process in the positive quadrant with underlying measure lebesgue measure, and describe a path g(t) through the plane where g(t) is upper rh corner of rectangle of area t and t rectangle is not s<t rectangle plus disjoint piece. – mike Apr 17 '12 at 11:40
(+1) Thanks for the edits! :) – cardinal Apr 17 '12 at 14:07
Should "counting process" also mean it is monotone increasing and right-continuous? Otherwise you could let all the $N(t)$ be independent. – Nate Eldredge Apr 17 '12 at 16:12
up vote 4 down vote accepted

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 $m>s$.

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I'm curious if there is any construction with a fixed parameter for $s$ like in the above description. That is can we make $Z_t$ a function of $t$ alone? – Srinivas Eswar Nov 6 '15 at 3:24

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!

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