This is all true, but these are all points I make in the post already.
Edit: I couldn’t point this out earlier because I was on my phone, but the maximum attained by a random walk is not actually O(√N), it’s O(√NloglogN). It really doesn’t affect your argument, but strictly speaking the claim is otherwise not true.
Edit: I couldn’t point this out earlier because I was on my phone, but the maximum attained by a random walk is not actually O(√N), it’s O(√NloglogN). It really doesn’t affect your argument, but strictly speaking the claim is otherwise not true.
Interesting. You were correct and I was wrong here. I was unaware that the expected maximum was enough larger than the expected final value that it ended up asymptotically different.
This is all true, but these are all points I make in the post already.
Edit: I couldn’t point this out earlier because I was on my phone, but the maximum attained by a random walk is not actually O(√N), it’s O(√NloglogN). It really doesn’t affect your argument, but strictly speaking the claim is otherwise not true.
Interesting. You were correct and I was wrong here. I was unaware that the expected maximum was enough larger than the expected final value that it ended up asymptotically different.
Is it?
Ctrl-F “random walk” in the post shows no hits (aside from this comment.)
I don’t use the same vocabulary but your profit from the game is obviously a random walk.
It is obvious in hindsight; many things that are obvious in hindsight aren’t necessarily obvious until pointed out.
I thought it good to explicitly point out the connection in case someone didn’t see it.