Can I make a Poisson process by combining several non-Poisson processes?

Question

I'm interested to know if I can split a Poisson process into non-Poissonian sub-processes or not. Or equivalently, I want to know if an (ensemble) Poisson process can be produced by other-none Poissonian processes or not.

Note:

• I don't care about the number of processes, as long as it remains countable.
• A simple (counter) example that shows the (lack of) existence of such sub-processes is enough for me, although, of course, more advanced resources are very welcome.

Answer 1

This summarizes my comments and gives another example. Define $N(t)$ as a Poisson process with parameter $\lambda>0$.

1. If you do not require the sub-processes to be independent, you can take $$N(t) = N(t)/2 + N(t)/2$$ and $N(t)/2$ is not Poisson since it can take non-integer values. You can also take anything, like $Y(t) = \min[N(t), 2]$ and $Z(t) = N(t) - Y(t)$, and then $N(t) =Y(t) + Z(t)$.

2. If you want $Y(t)$ and $Z(t)$ to be independent, you can take $Y(t) = -5$ and $Z(t) = N(t) + 5$.

3. For a more interesting example with $Y(t)$ and $Z(t)$ independent, you can take $Y(t)$ as the accumulated arrivals over $[0,1]$ and $Z(t)$ the accumulated arrivals over $(1, \infty)$, and $N(t) = Y(t) + Z(t)$:

\begin{align} Y(t) &= \left\{\begin{array}{cc} N(t) & \mbox{ if $t \leq 1$} \\ N(1) & \mbox{ else}\end{array}\right. \\ Z(t) &=\left\{\begin{array}{cc} 0 & \mbox{ if $t \leq 1$} \\ N(t)-N(1) & \mbox{ else}\end{array}\right. \\ \end{align}