LED light turns on

It could be one of a few things. As has been noted by @JRaefield, it could be a leaking dimmer or touch switch. Many dimmers designed for incandescent bulbs do leak a little current which is imperceptible in incandescents but this current is enough to occasionally cause LED bulbs to glow. I've read (don't have references, sorry) that some old "smarter" touch switches and dimmers actually work on a the principle that they leak a small residual current while "off" in order to power themselves. This only becomes a problem when your LED bulbs require comparatively little current to start lighting up.

Another possibility if you don't have a dimmer is a capacitive loop where the switch is located on the neutral instead of active. See this answer for details. Basically the wire from the (open) switch to the LED lamp is one plate of a large capacitor, and earth is the other plate. A small amount of AC traverses the "capacitor" completing the circuit. Getting the switch re-wired may help.