Key word being eventually
When you can only cross a barrier in one direction, even if movement is random and weighted away from the barrier, statistically everyone ends up on the other side of the barrier after a certain amount of time (granted, it might tend to infinity). Especially in this case, since you can't get arbitrarily far away from the barrier.
As long as the rate of promotions is sustained by the arrival of new players, this shouldn't be a problem. But well, it depends on both rates...
that's actually not true. if you have 50% or more for a win you will reach it eventually, but if you have less than 50%, there is a real percentage chance that you will reach it, and likewise a chance that you won't.
unless there is a gap on how low you can drop (like 0% progress in this case), then you do have 100% with infinite amount of time.
I was waiting to see how long it would take for someone to correct me... yeah, you are right, I spoke too soon. To prove that it works in this specific case, you would need to modelize the exact win probability and related rating increase. Note that because rating doesn't change by discrete amounts (I mean, technically yes, the precision of their floats), we can't really talk about rating being bounded from below.
But still, given the way ratings seem to work, and that your win chance shouldn't ever be much lower than 0.5, I think this holds.
If your odds of winning don't change based on your position (i.e. always 40%) then you are guaranteed to promote in the infinite time limit. This is like the Gambler's ruin only promotion is going "bankrupt". Here's a link about it: http://www.statslab.cam.ac.uk/~james/Markov/s13.pdf (on pg. 3)
This does not require a cap on how low your position can drop (or on how much money the gambler can win). There will still be a sequence of events which you are guaranteed to encounter in the infinite time limit that will lead to promotion (or bankruptcy).
After they talk about the gambler's ruin there's an example where you aren't guaranteed to end up in the absorbing state (which for us is promotion), it requires the probabilities to change based on your position. I didn't look at it closely but I'm pretty sure it will require (at some positions) your odds of winning to decrease the further you go down in position which doesn't seem like a very good model.
My guess is that to escape the absorbing state you need to have accelerating probabilities away from that state. I can't imagine having "stable" regions away from the absorbing state will cut it as infinite time will get you there for sure. I don't really know anything about this stuff though.
Your link does state that the odds of getting into the absorbing state are less than 1 if your odds of moving away from the absorbing state are strictly higher than 0.5. You don't need "stable" regions away from the absorbing state.
You can think of it this way: you start in some random position x>0. Because the chances of getting away from the absorbing state (+1) are higher than those of approaching it (-1), eventually all paths (ignoring the absorbing state 0 for the time being) will go to +infinity. So now you just have to "count" how many of them pass through zero, and how many don't. So you've got a non-zero chance to pick a path that will never cross zero.
On the other hand, if probabilities were the other way, then all paths would go to -infinity, which means all paths cross zero at some point.
If you've got a 50/50 chance to win (+1), it's a bit harder to conceptualize the maths...
Maybe blueblimp wants to start a new thread where he works through the maths of ELOs and tiers?