Michael Kleber: So to bring things back closer to David's original question, these constructions prove that the probability of the current rightmost bullet getting replaced eventually is nonzero. Can anyone prove that that probability is 1? Or might there be some configuration of bullets in which we expect the current rightmost bullet to survive forever? --Warren Smith: well, the trouble is, the probability certainly is nonzero and my construction as slightly corrected by Wechler shows that. But it might be that the sum of all replacement probabilities over all states in the whole infinite future, i.e. the expected total number of replacements, is 0.001... But now this argument occurs to me: 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. This also shows the position of the rightmost is NOT asymptotic to t. CASE B: On the other hand, if we know the total number of leader-replacements stays finite almost surely, then it seems to me we'd also know that the rightmost guy's speed does not tend to 1, but rather to some V<1 which after some amount of time just stays constant forever after. Then the leader position is asymptotic to V*t, not t. CASE C: With some probability P (0<P<1) the number of replacements is infinite, but with probability 1-P it stays finite. Then it seems to me you can argue using a mixture of the A and B arguments. CONCLUSION: So this three-pronged argument seems to me to settle Wilson's question: Does the leader-speed tend to 1 with time t? Does his position asymptote to t? Answers: "no" and "no." However, this ultimately seemed a little too easy, so maybe I screwed up; certainly I've left the details kind of loose. Assuming I did not screw up, and this really is that easy, then actually this "easy" argument might be generalizable to have pretty wide applicability to proving way more stuff, and it'd be nice to figure out what that is.