Since each "sum" of the time series you have is Independent and Normally distributed, you need a different approach from autocorrelated time series models:

(1) http://en.wikipedia.org/wiki/Prediction_interval 

(2) https://stats.stackexchange.com/questions/25055/how-to-calculate-the-confidence-interval-for-time-series-prediction

I leave it for the experts here to illustrate how to go about with methodology(1) as I have no idea of the detailed steps to construct Prediction Intervals for your discrete time series.