I have had the exact same problem, and mine was also a 2011 Jensen model. For a 2 month period earlier this spring it would randomly turn itself on parked outside my house. Not near other houses or trailers with remotes, or Indian burial grounds :B
It also wasn't the alarm, I checked, and the day and times of day it turned on was random.
One thing I did that stopped it once we moved the trailer and took it camping this summer; I cleaned the battery terminals as they were a little corroded. I dont think it was related, but it hasn't happened since.... I suspect it is faulty manufacturing/wiring in the radio unit...