I understand how to prove that the sum of $n$ independent Poisson random variables is a poisson random variable mathematically. However, I don't understand this in terms of the concept of Poisson processes.
A poisson process must have accidents occurring at a constant rate, independently and singly. I understand that in the combined process accidents will occur independently and at a rate of the sum of the $n$ different rates but I'm not sure how accidents are guaranteed to occur singly in the combined process. E.g. below, how do you know a white and black dot won't overlap in the merged process?
Also, if someone can explain how to deduce the rate of the combined process from the concept itself instead of the mathematical proof that would be great.