So. I've acquired the unenviable task of having to learn renewal theory on my own. I'm finding most of it to be pretty intuitive, except for one thing. The intuition behind the renewal equation has completely eluded me.
Every resource I've found so far basically dives right in with something like:
If $f$ is a function in $[0,\infty)$ bounded on finite intervals and $F$ is a measure on $[0,\infty)$, an equation of the type $Z = f + Z \ast F$ is called a renewal equation. If $f$ is directly Riemann integrable, then the solution to this equation is given by... If $F(\infty) < 1$ then the equation is defective and if $F(\infty) > 1$ then the equation is excessive... etc...
But... where does this equation even come from and why is it so important? It's just one possible relation out of many. Is there a natural, simple example to see how it arises in a basic, pure renewal process? I'm having a pretty hard time finding a textbook which does a good job of this. For example, I can derive that if I have a renewal process with renewal function $m(t)$ and interarrival distribution $F$, then
$$ m = F + m \ast F $$
But when I move to more complicated functions besides $F$, I'm no longer sure what the result is saying. For example in the above example, the general solution for renewal equations says that $m$ also has a convolution form involving the measure $\sum_{n=1}^{\infty} F^n$, where $F^n$ is the distribution of the $n$th renewal time. I can see that this measure is just the renewal function, but I'm having trouble interpreting it.
I know it's asking a lot, but can someone de-mystify some of this? If it helps at all, I have a pretty decent grasp of measure-theoretic probability (but not good enough, apparently) and I'm trying to follow Asmussen's chapter on renewal theory.