Swapping Poisson process at fixed time

Question

Let $X(t)$ and $X'(t)$ be independent Poisson processes with rate $\lambda$. Fix a time $s\in \mathbb{R}$. Define $$X''(t)=\begin{cases} X(t) & t\leq s, & \\ X'(t)-X'(s)+X(s) & t>s. \end{cases}$$ Is $X''(t)$ also a Poisson process with rate $\lambda$? I suspect it should be from the memoryless property of the exponential, though I can't seem to find this question in the literature and am unsure on how to prove it. I've seen similar questions where the switch occurs after some number of arrivals, but not at a deterministic time.

Answer 1

More generally, if $X(t)$ has rate $\lambda_1$ and $X'(t)$ has rate $\lambda_2$, then $X''(t)$ is a non-homogeneous Poisson process with rate step function $$\lambda(t) = \begin{cases} \lambda_1, & t \le s \\ \lambda_2, & t > s. \end{cases}$$ But in your case, if $\lambda_1 = \lambda_2$, then $\lambda(t) = \lambda$ is constant and therefore $X''(t)$ is homogeneous. That the switching time is deterministic has no bearing because the rate has not changed.