Question

Exercise 1. Suppose (Nt):20 is a Poisson process of rute λ uith respect to the First Definition (Srnall interval properties). For each n 2 0 and t E [O,oo) define Pn(t) := P(N, = n). Suppose t>0 and h < 0 is such that lht. Show the following two things: (1) Po(t +h) - Po(t)AhPot)(h) as h-0, and (2) for each n 2 1, P(t+h)-P(t)-hh) P(t)+(Ah+o(h))P-)+o(h) as h Note: the proof in the notes is only done for h >0, here you are supplying the argument for the case h < 0 so you need to be careful. Recall: here each function o(h) represents some function with the property lim0(h)/h-0 These functions do not have to all be the same and we do not care what they are.

Answer 1

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