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It does sound like the PSU is dying. If you have a PSU tester, check the voltages of 3.3v, 5v and 12v rail, and see if they are off by more than 5%. When my old PSU died, the computer restarted a few times, and then it wouldn't turn on any more.
bad memory sticks wouldn't shut the pc down, the memory either won't recognise any more like it wasn't plugged in (completely dead), give a bsod error code on loading or the bios will alert you when it runs the mem test on start up (assuming you have the speaker plugged into the mobo).
if you only had one, it would probably just bsod and restart and then the bios would alert you and that would be it.
overheating graphics cards can do a hard shut down of a computer.
Bad ram has caused me blue screens, unless blue screens were turned off and the PC would restart instead. No bios alert, windows would start fine. It depends on the extent of the damage to the RAM.
It can progress from causing problems under load, to simple things like trying to browse the web.
Keep in mind I'm not insisting it IS the RAM, just throwing it out there as something to check.