A poisson process $N(t)$ is a Levy process, where where $N(t)-N(s), t-s=h$ has a discrete distribution $(\lambda h)^k e^{-\lambda h}/k!$.
I want to prove that the jumps of this process has value 1 a. s. I am not ble to prove it if we have the entire real line, only if we work on the interval $[0,T]$. Can you please help me expand the proof to the entire real line?
For $[0,T]$ I have: Call the event that a jump with value bigger than 1 happens A. The complement of this event $A^c$ is that all jumps are 1. Partition the interval $[0,T]$ in increments of length $T/n$ for arbitrary n. Then $A^c$ contains in the event that for each increment in our partition the increment value of our process is either 0 or 1,call this event B$_n$. That is, we have $B_n \subset A^c, \forall n$ . If we can prove that the probability of $B_n$ converges to 1, we must have that the probability of A is 0. But $P(B_n)=(e^{-\lambda T/n}+e^{-\lambda T /n}(\lambda T/n))^n$, but this expression converges to 1 as n tends to infinity.
However, how do I modify the proof when we have the entire real line? I don't see how to get the proof to work then?
