Cautiously approach a downed big game animal from the rear and determine if it is still alive. Watch carefully for any rise and fall of the chest cavity. Be aware if the animal's eyes are closed. The eyes of a dead animal are usually open. If the eyes are open and glazed over, the animal is likely dead.
If your quarry is still alive the animal should be dispatched in a quick and humane fashion. For gun hunters a single shot to the base of the ear will suffice. If the animal is to be mounted, however, place a single shot into the heart/lung area. For bowhunters, a singe arrow into the heart/lung area is the only choice.
A responsible hunter will not deliberately create additional stress on a dying animal.