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Chapter 1 Practice Question 4 ii)

A

Alina Miclea

Member
Hello,
The question 4 point ii) asks when a compound Poisson process is also a Poisson process.
The answer is that this is satisfied when the increments only take the values 1 or 0. However, this answer is far from obvious to me. Why it is not sufficient for increments to have a Poisson distribution?
Could someone explain it to me?
Thanks!
 
It might be helpful to consider the transition diagram for a Poisson process (see Ch4 p27). This diagram illustrates the fact that events must occur singly in a Poisson process, ie we must go from state 0 to state 1 to state 2, etc. If Xj has Poisson distribution, then it can take any non-negative value. This would mean that jumps would be possible to non-adjacent states, so X1 + X2 + ... + X_N(t) would not be a Poisson process.
 
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