HVAC Fan Running

We haven't used our AC for 3 weeks. When I got up this morning, I noticed the fan running. The thermostat is the typical programmable battery powered box store thing that is about 15 or 20 years old.
I don't know much about these systems, but I would imagine if the unit had been running, and the fan didn't shut off, it would be a limit switch or relay or?
But what would cause the fan to start running by itself?
I cycled the stat through the heat, ac and off again. No luck.

I just took the batteries out and it stopped. Is it the stat?