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I have the following definition:

Definition (Poisson process). $N(t)$ is a poisson process if $N(t+h)-N(t)$ and $N(t)-N(t-k)$ are independent and $N(t+h)-N(t)\sim N(h)-N(0)$.

We try to find the law of such a process. In my course it's written that since $\mathbb P\{N(t)-N(t)=1\}=0$, we have that $$\mathbb P\{N(t+\Delta t)-N(t)=1\}=0+\lambda\Delta t+o(\Delta t).$$

I know it's the first order of taylor serie, but I don't understand how we can get it. What is the function ? (I guess it's $f(x)=\mathbb P\{N(t+x)-N(t)=1\}$, but what would be the derivative of such a function ?)

In the same way, how do we get $$\mathbb P\{N(t+\Delta t)-N(t)\geq 2\}=o(\Delta t)$$ and $$\mathbb P\{N(t+\Delta t)-N(t)=0\}=1-\lambda\Delta t+o(\Delta t)\ ?$$

idm
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  • This is not the definition of a Poisson process, only necessary conditions. – Did Aug 06 '16 at 10:45
  • It's my definition. Page 6 (if you read french : https://www.agroparistech.fr/IMG/pdf/PoissonNaissancesEtMorts-AgroParisTech.pdf @Did – idm Aug 06 '16 at 11:02
  • You are lacking at least the conditions that every $N(t)$ is integer valued, that $(N(t))$ is non decreasing and that $P(N(t+h)-N(t)\geqslant2)=o(h)$ when $h\to0+$, which in no way can be deduced from this "definition". Consider for example $(N(t))$ Poisson and $X(t)=2N(t)$, does $X$ satisfy the conditions in this "defnition"? Is $X$ Poisson? Ergo. (Unrelated (or, very related?): I love how every trivial remark made à propos your questions must be justified ad nauseam before you accept to consider there might be some grain of truth in it...) – Did Aug 06 '16 at 11:17

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