Here are two solutions which are the best ways to deal with the blank display issue on your Samsung Galaxy Note 2:
1. Connect the charger to your device and let it charge for a few minutes.
Turn on the device by pressing and holding the power button simultaneously.
If the problem still exists, try this:
2. Remove the battery for 30 seconds with the device still turned on. After a minute or two, replace the battery Power on the device.