Clarifying the following concepts: renewal processes, stationary processes, counting processes and point processes

Question

Is it true that a renewal process is a stationary process on $\mathbb{R}^+$ and a counting process is a point process on $\mathbb{R}^+$?

So the latter concepts are the generalization of the former ones?

Answer 1

Renewal processes are a class of stochastic processes (including the Poisson process), that have all the properties of the Poisson process, except the condition that the inter-jump-distribution is Exponential. I.e. Renewal processes start at 0, jump by 1 after waiting a random length of time $T$. The waiting time for each subsequent jump is independent of the evolution of the process up to the time and position of the last jump, i.e., the process starts anew from the last position.

Point processes are a further generalization. Roughly speaking, the point process is like a renewal process, except the jumps need not be of size 1.

As for stationarity, you need to be a bit more careful phrasing it. Poisson processes and renewal processes are not stationary, but their increments over time intervals of fixed lengths are. For instances, if your Poisson process is $N_t(\lambda) = N_t$, with $\mathbf{P}(N_t = i) = \frac{\lambda^i e^{-\lambda}}{i!}$, then $N_s$ and $N_t$ don't have the same distribution if $s \neq t$, but for $r > 0$, $N_{t+r}-N_t$ and $N_{s+r}-N_s$ have the same distribution, since both random variables are increments of the process over intervals of length $r$.