On Mon, Jun 11, 2012 at 7:34 PM, Warren Smith <warren.wds@gmail.com> wrote:
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.
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. 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.) --Michael -- Forewarned is worth an octopus in the bush.