re: Can a tv just turn on by itself?

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My parents had a tv that would turn off on its own. So I see no reason why one couldn't turn itself on.

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that's differnt. Power supply failure is the most common problem for TVs.
When some are on for a bit they heat up and the contacts swell and you get an interruption of power. You will still see standby power as that LED is fed from the power input of the cord. But power to the TV control board will be cut.
Most times it will then just turn right back on.
But when this starts happening change the board with the fuse on it in your TV.