Intuitively, I think that it is possible for a sum of time series to be white noise, when the individual time series are not. Reason I am asking, is because I want to know if it's useful to investigate if it's better to model the individual time series than the sum of the time series. In my case, each of these time series are the sales of the same product, but sold at different locations (hence multiple time series). Please note: in the end I am interested in forecasting the sales of the joint time series, and not the individual ones.
For the other way around, I found an answer here. In brief: the sum of two independent white noises, results in a white noise process.