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I have a question when reading Essentials of Stochastic Process by Richard Durrett, 2.2.1 Constructing the Poisson Process. It says, Let $\tau_1,\tau_2,\dotsc$ be independent exponential$(\lambda)$ random variables. Let $T_n=\tau_1+\dotsc+\tau_n$ for $n \geq 1, T_0=0$, and define $N(s) = max \{ n:T_n \leq s \}$.

To show that $N(t)$ is a Poisson process, the book first shows $N(s)$ has a Poisson distribution with mean $\lambda s$. To show that $N(t)$ has independent increments, the book proves this Lemma: $N(t+s) -N(s), t\geq 0$ is a rate $\lambda$ Poisson process and independent of $N(r), 0 \leq r \leq s.$ Here it has this statement that I can't wrap my head around, "... the interarrival times after s are independent exponential$(\lambda)$, and hence that $N(t+s) -N(s), t\geq 0$ is a Poisson process." screenshot of the page

It confuses me because the whole section is trying to show that $N(t)$ is a Poisson process via the definition, (i) $N(0) = 0$, (ii)$N(t+s) -N(s)=Poisson(\lambda t)$, and (iii) $N(t)$ has independent increments. But when proving independent increments, it just proves that $N(t+s) -N(s), t\geq 0$ is a Poisson process. It seems that "the interarrival times are independent exponential$(\lambda)$" is sufficient for $N(t+s) -N(s), t\geq 0$ to be a Poisson process, but then by the same logic, isn't $N(t)$ automatically a Poisson process?

If anyone can explain to me why is $N(t+s) -N(s), t\geq 0$ a Poisson process, or why are the increments independent, that would be much appreciated.

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1 Answer 1

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In the confusing statement the author meant as the Poisson process the object constructed in your first paragraph (i.e. the process constructed using exponential RVs). Note that this is what he actually proves. It makes sense to do that, because then the lemma gives that $N(t+s) - N(s)$ has the appropriate Poisson distribution and is independent of $N(r)$ for $0\leq r\leq s$.

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