As the program runs for a while via a scheduled task without any apparent problems, the indication is that there may be a problem either with the task setup, or there is a bug in Task Scheduler itself.
If it was a program problem, it probably wouldn't run at all or it would start to run & generate an error code.
There was a bug in Vista in that user generated scheduled tasks could sometimes stop running for no apparent reason.
This was fixed in Windows 7, but it may just be possible that Windows 8 is also affected.
It may just be worth testing by creating a batch file containing just this detail:-
echo %time% >> C:\Utilities\HG612_Modem_Stats\Scripts\Schedule_Test.TXT
Then create a scheduled task to run the batch file every minute, using identical details to the stats task.
If that also stops running after a few hours, we could probably rule out a program problem, then start looking at Task Scheduler itself for an answer.