I downloaded an offline map that covers the area from my home to my place of work 20 miles away, then put the Nexus 4 into airplane mode.
Went out my front door, got into the car, plugged the phone into the charger. Started Google maps and cancelled out of the request to turn on a data connection, then loaded my offline map. The small blue arrow immediately showed my current location on the map.
As I drove to work, the blue arrow accurately tracked my position on the map. One thing I did notice: with a "live" map, the map automatically moves such that the blue arrow remains in the middle of the screen. However with the offline map, the map remains stationary while the blue arrow moves, eventually moving all the way off screen. At that point, I can either drag the map into position or tap the icon at the top right of the screen to automatically move the arrow back to center screen and center the map. During all this time, the GPS icon was present at the top left of the notification bar.