Let $X_t$ be a homogeneous Poisson process of rate $\lambda$. Suppose we define functions $p_1(t)$, ..., $p_k(t)$, such that for all $i$ and $t$, $p_i(t)\in [0,1]$ and $\sum\limits_{i=1}^kp_i(t) = 1$.
Let $Y_1,\dots, Y_k$ be processes. Now, for each arrival to the process $X_t$, choose randomly one of the processes $Y_1,\dots,Y_k$ according to the probabilities $p_1(t),\dots,p_k(t)$ and let this arrival be an arrival into the chosen process.
Does this produce independent, inhomogeneous Poisson processes and if so, how to prove it?
Considering the infinitesimal characterization of a Poisson process, it seems likely, but I'm not really sure where to start.
Thank you.