I'm currently trying to understand Poisson Processes using this material: https://www.open.edu/openlearn/pluginfile.php/1118700/mod_resource/content/3/Modelling%20events%20in%20time_m343_1.pdf (this question is based on pages 29 and 30).
Say we have three types of events $A$, $B$, and $C$ that occur independently with rates $\lambda_A$, $\lambda_B$, and $\lambda_C$. According to the document, this is a multivariate Poisson Process which consists of three independent univariate Poisson processes with rates $\lambda_A$, $\lambda_B$, and $\lambda_C$.
Now I'm wondering if this can also be modeled as follows (inspired by the description on the bottom of page 30): Take a univariate Poisson Process with rate $\lambda = \lambda_A + \lambda_B + \lambda_C$. Everytime an event occurs, we determine its type $t$ by drawing from the discrete probability distribution $T$ with the following PMF: $$ p_T(t)= \begin{cases} p_A=\lambda_A/\lambda, & t=A \\ p_B=\lambda_B/\lambda, & t=B \\ p_C=\lambda_C/\lambda, & t=C \\ 0, & \text{otherwise} \end{cases} $$ Is this equivalent? Because I'm not sure if this hurts the independence assumption or changes something like the expected amount of events of a certain type in a certain interval.
Edit: If it is, would it also be equivalent if the univariate Poisson process of each type $t$ would be non-homogeneous, i.e. $\lambda_t$ would be a function of time instead of a constant?