In the book Probability and Random Processes by Grimmett and Stirzaker, a Poisson process is defined to be any process $(N(t))_{t\in[0,\infty)}$ with values in $\mathbb{N}_0$ satisfying following three properties:
$\hspace{20pt}$(a) $N(0) = 0$ and $s
$\hspace{20pt}$(b) $\mathbb{P}\left(N(t+h)=n+m|\hbox{ }N(t) =n\right) =\begin{cases} o(h) & \text{if } m>1,\\ \lambda h + o(h) &\text{if } m=1,\\ 1 - \lambda h + o(h) & \text{if }m=0\\ \end{cases}$
$\hspace{20pt}$(c) if $s
I am having problems with understanding the meaning $o(h)$ in (b), so my question is:
How do we interpret $o(h)$ in this definition?
I know $o(h)$ in the classical sense is supposed to be just a function such that $\lim_{h\to 0}\frac{o(h)}h=0$. But here, $o$ is used three times and I suspect it has a different meaning each time. (Since otherwise, we would necessarily have $o\equiv0$.)
So, I did some thinking and reinterpreted (b) to read:
$\hspace{20pt}$(b') There exist functions $o_1, o_2, o_3$, such that for $i=1,2,3$:$\lim_{h\to 0}\frac{o_i(h)}h=0$ $\hspace{38pt}$and $\mathbb{P}\left(N(t+h)=n+m|\hbox{ }N(t) =n\right) =\begin{cases} o_1(h) & \text{if } m>1,\\ \lambda h + o_2(h) &\text{if } m=1,\\ 1 - \lambda h + o_3(h) & \text{if }m=0.\\ \end{cases}$
But I am completely confused about the order of quantifiers in this statement (which is why I left this rewording a bit vague). Should the functions $o_1,o_2,o_3$ be the same for all $t,n$ and $m>1$? (That is: should we take a separate function $o_m$ for each m? Or even a separate function $o_{t,n,m}$ for each triple $(t,n,m)$?)
Does this even matter or are these definitions miraculously equivalent?
I would really like to know what exactly it is this definition is trying to define. Thanks in advance for any helpful suggestions.