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In my notes I have a definition for $o(\Delta)$ which states that if a function $f$ is $o(\Delta)$ then as $\Delta$ approaches zero, $f(\Delta)/\Delta = 0$.

Then the notation gets used in equations such as this:

$$P({N(t+\Delta) - N(t) = 1}) = \lambda*\Delta + o(\Delta),$$

where $P$ stands for probability and $N(t)$ is a number of events that has occurred up to time $t$ (Poisson process of parameter $\lambda$)

Can someone please explain what the significance of this function is, and how it is getting used in that equation as a variable, when the definition states that it simply describes a function?

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  • $\begingroup$ Little-o notation. $\endgroup$ – Did Mar 29 '12 at 6:51
  • $\begingroup$ I reformatted your question and chose $\Delta$ for your delta. Is this OK? $\endgroup$ – draks ... Apr 4 '12 at 9:27
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This is the "little-oh" notation. If $f$ and $g$ are functions and $f(x)/g(x)\to0$ as $x\to0$ we say $f(x)=o(g(x))$. An example of this is $f(\delta)=o(\delta)$ means $f(\delta)/\delta\to0$ as $\delta\to0$, as you have seen.

$P(\{{N(t+\delta)-N(t)=1\}})=\lambda\delta+o(\delta)$ just means that the left side is $\lambda\delta$ plus some function that goes to zero faster than $\delta$ goes to zero.

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