What make is it?.
If you suspect PSU, I would run the scope round the supply rails (on the outputs of the rectifiers), to check for ripple caused by high ESR electrolytics.
The dim display 'may' be a red herring though, as (assuming it's a CCFL display) they tend to have a fairly short lifespan, and rapidly become too dull to see.
Very, very, occasionally you do find a duff PSU capacitor causing it though - I've changed dozens of high ESR capacitors for dim displays, and only improved things in a couple of cases (and not great then, but better than it was).
It 'could' be shutting down due to an output protection fault, due to a duff output IC, or (more commonly in surround systems) dry joints on the IC's or surrounding components.