I'm learning about renewal theory at the moment and I'm getting a little confused because aren't they practically the same thing - I don't understand the point of having a distinction between "renewal" and "regenerative" processes, especially because from what I understand renewal processes are regenerative anyway.
Can someone provide a high level explanation of what the actual differences are?
A renewal process is a sequence (a discrete-time process) of random times, $0\leq T_0<T_1<\ldots<T_n<\ldots$ such that the increments $T_1-T_0$, $T_2-T_1$ are i.i.d. variables independent of $T_0$.
A regenerative process $\{X(t),t\in \mathcal T\}$ is a stochastic process (either discrete-time with $\mathcal T=\mathbb N$, or contunuous-time with $\mathcal T=R_{+}$) possessing a renewal process of regeneration times. THe regeneration times split a process path into stochastically independent and identically distributed 'tours' $\{X(t), T_{i}\leq t< T_{i+1}\}$, $i=0$, $1$, $\ldots$. We want the regeneration times to form a renewal process because we want the tour durations to have the same probability law and to be mutually independent.
You can construct a simple regenerative process from a renewal process by taking 'forward residual times'. Let $N(t)=\min\{n\colon S_n> t\}$ be the number of renewals in $[0,t)$ and set $X(t)=t-T_{N(t)}$. That's it. Then the renewal times $T_n$ will be at the same time regeneration times for the regeneretive process $\{X(t),t\geq 0\}$.