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) 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.