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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

1 Answer 1

up vote 3 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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