If $N(t)$ is a Poisson process with parameter $\lambda(t)$ then is $N'(t)=N(t+2)-N(2)$ a poisson process? I think it should be poisson process as it is like observing a poisson process after time $2$ but how to prove it.
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Hint: A Process $N$ is a poisson process with intensity $\lambda$ iff:
- $N(0) = 0$ almost surely.
- $N(t) - N(s) \sim \mathrm{Poi}(\lambda(t-s))$ for all $0 \le s < t$.
- $N(t) - N(s)$ is independent of $\sigma(\{N(k) \mid 0 \le k \le s\})$ for all $0 \le s < t$.
- $N$ has almost surely càdlàg paths.
Dominik
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For OP's benefit, condition 4 should be equivalent to $P(N(t+h)-N(t)>1)=o(h)$. – Alex R. Sep 01 '15 at 18:55
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@AlexR. I don't see why this makes the proof easier, it's pretty obvious from the definition that $N'$ holds the last property if $N$ holds it. – Dominik Sep 01 '15 at 18:59
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What is càdlàg paths? – Subhasish Basak Sep 01 '15 at 19:13
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càdlàg means right continuous with left limits. – Dominik Sep 01 '15 at 19:16