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.