I'm wondering if there's ground radar on planes, or any fallback system other than relying on a controller, for avoiding runway incursions?
Have any studies or reports been made that show statistically how the majority of these incidents occur?
At least three of the "fallback systems" I know of are looking out the window, being familiar with the airport and its markings (as well as associated hotspots, as airport diagrams may indicate), and also paying attention to the radio, if you're equipped. Of course, no one of these methods is foolproof, so it takes a combination of alertness and prudence to avoid incursions. I was taught to look both ways before intersections, or to look both ways before entering a runway, just to make sure I didn't miss anything. Also, using lighting on your aircraft appropriately helps make you more visible.
Other planes may not have a radio, so you may not hear a radio call of someone else in the pattern. Also, at a complex intersection in a new airport, it may be confusing where you are, or what hold short lines you're about to cross -- so being aware of your surroundings is important here.
There are some technologies the FAA has been implementing to reduce the risk of runway incursions, such as Elevated Runway Guard Lights and stop bar lights, plus ground radar technologies (for ATC) to assist (see ASDE-X and AMASS). Note that your A/FD entry may list, for example, that not all taxiways are visible from the tower -- which is a good reason to stay extra alert.
An additional FAA training video:
There are many procedures and systems in place to reduce the risk of runway incursions as much as possible. Here's an outline of some of the areas where these are being applied:
According to Eurocontrol, the majority of runway incursions happen due to communication breakdown, ground navigation errors due to inadequate or ambiguous surface signs and markings and relevant information needed in the cockpit, and occur during taxiing out and departure operations.