check this wire (
http://www.automotiveforums.com/vbul...4&d=1269150248 ) on the firewall side of the engine (crank pos sensor). wire burns easily on the exhaust and melts insulation and shorts out the sensor. not sure if this applies to your year lumi. you can see in the picture the plastic loom was burned in half, and burned the insulation on the wires which I replaced long before taking this pic) Up until I replaced that wire a few years ago, it would run, then completely die, then after cranking would sometimes start up... I guess the rocking of the engine during starting would move the wire and sometimes un-short it.
aside from the sensor, i'd see if you dont have some kind of electrical problem, charging problem, or faulty battery....