Before marking this post as a \"duplicate\", I am writing this post because no other post holds the solution to the problem.
I am trying to turn off the device, then
I wouldn't have hope of "waking the screen" in the activity. If the screen is off the activity is probably in a paused state and shouldn't be running any code.
When waking up, there is the issue of the lockscreen. I don't know how any app can automatically bypass the lockscreen.
You should consider running your background tasks in a service, and then using the notification manager to send a notification when whatever is detected. The notification should provide some sort of device alert (screen wake up, notification icon, notification led, etc). When clicking the notification it can launch the intent to start your activity.
You could also attempt to start the activity direct from the service, but I really don't know if that will turn the screen on or bypass the lockscreen.