Is the sum of several Poisson processes a Poisson process? I would like to write an application that emits events distributed by a Poisson process with some $\lambda$
I need to separate the generation of these events in multiple processes, but I only have a $\lambda$, so I need to come up with a proper $\lambda_p$ for each process.
Is the sum of $p$ poisson processes, each with $\lambda_p = \frac{\lambda}{p}$ a Poisson process with parameter $\lambda$?
 A: If they're independent of each other, yes.
Indeed a more general result is that if there are $k$ independent Poisson processes with rate $\lambda_i, \, i=1,2,\ldots,k$, then the combined process (the superposition of the component processes) is a Poisson process with rate $\sum_i\lambda_i$.
It's really only necessary to show the result for $k=2$ since that result can be applied recursively. 
One way is to show that the conditions for a process to be a Poisson process are satisfied by the superposition of two Poisson processes.
For example, if we take the definition here, then the properties of a Poisson process are satisfied by the superposition of two processes: 


*

*N(0) = 0  $\quad$ (clearly satisfied if it's true for the components)  

*Independent increments (the numbers of occurrences counted in disjoint intervals are independent of each other) $\quad$ (follows from the independence mentioned above)  

*Stationary increments (the probability distribution of the number of occurrences counted in any time interval only depends on the length of the interval) $\quad$ (if it applies to the independent components it will apply to their superposition)    

*The probability distribution of N(t) is a Poisson distribution $\quad$ (see here*)

*No counted occurrences are simultaneous $\quad$ (simultaneity is an event with probability 0: follows from continuity and independence)  


* (though I'd regard this as a consequence of the other properties)  
