Unfortunately, my knowledge of the inner workings of Cinema Displays is very small, so I'm sorry I can't help a fellow Cinema Display user any further. Someone else with more knowledge here?
From my experience, whenever stuff suddenly powers on / off it usually pointed towards some capacitator problems. If it's sitting there dead now there's no point in it, but before I was wondering if the USB ports on the back were still supplied with power. Though if it's dead now, there's no point in it. Are there any good repair manuals for ADCs? I have never checked ...