CASE A: *IF* we knew that probability-sum approached infinity, or more precisely that the total number of leader-replacements -->infinity almost surely (which follows from its expectation being infinite), then we'd know every leader gets replaced eventually, hence could deduce that the ultimate speed of the rightmost guy does NOT tend to 1, since no matter how far you go into the future, there are moments when the current leader is killed whereupon the situation is sort of wholy reset.
Michael Kleber: I don't believe that's correct. It could be the case that we expect the rightmost bullet to get replaced infinitely often, and also that the probability distribution of the speed of successive rightmost bullets tends to be weighted more and more heavily towards 1.
--WDS: The bullet that annihilates the leader, follows a line segment in the time-space plane. This line segment totally separates what came before & what follows. The new stuff is a total restart, totally independent of all the previous stuff. Hence the new speed probability distribution from then on, is the same as the old one starting at time=0.
Kleber: That is, I don't see any reason to believe that the PDF(velocity of bullet fired at time 10^3 that turns out to be rightmost) is the same as PDF(velocity of bullet fired at time 10^6 that turns out to be rightmost). (I talked about this a little in my first (excessively long) mail on the subject.)
--well, hopefully you now DO see a reason to believe a version of this.