# Splitting a Poisson process according to time-dependent probabilities

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.

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Indeed this is called splitting and the inverse operation is called merging. Yes a splitting produces independent Poisson processes with intensities $\lambda_i(\ )$, where $\lambda_i(t)=\lambda p_i(t)$, or $\lambda_i(t)=\lambda(t) p_i(t)$ if the original Poisson process is inhomogenous with intensity $\lambda(\ )$.
A proof for the homogeneous case where $\lambda(t)$ and the $p_i(t)$ are all independent on $t$ is there and the adaptation of this proof to your case is not difficult. The general case is in many classics on point processes, for instance in section 6.4 Marked point processes of An introduction to the theory of point processes by D.J. Daley and D. Vere-Jones.