Timeline for How to show a function converges to 1

9 events

when toggle format	what		by	license	comment
Jan 18 at 18:36	history	edited	Ash Malyshev	CC BY-SA 4.0	simplify messy computation
Jan 18 at 6:14	comment	added	Ash Malyshev		If I'm not mistaken, looking at $M = \log b + \log \log b$ and computing the probability that exactly one interval is stranded after $M$ gives you a bound $f(0, b) < 1 - (\log b)^{-1 + o(1)}$, so the probability goes to 1 fairly slowly.
Jan 18 at 4:25	history	edited	Ash Malyshev	CC BY-SA 4.0	deleted 2 characters in body
Jan 18 at 2:22	history	edited	Ash Malyshev	CC BY-SA 4.0	fix Omega notation.
Jan 18 at 1:44	comment	added	Ash Malyshev		Follow-up question: characterize which joint distributions of $X, Y$ have the property that sampling the distribution $n$ times and taking the union of the intervals $(X_i, Y_i)$ results in a contiguous interval with high probability as $n \to \infty$. Are there any nondegenerate cases where this isn't true?
Jan 18 at 1:10	comment	added	Ash Malyshev		I also suspect there's a less-tedious argument in the continuous-time independent-balls formulation. (E.g. something in terms of an explicit formula for the distribution of the ball-states at time $t$, which can be handled with matrix exponentials?) Alas I haven't the time to look for a shorter answer.
Jan 18 at 0:59	history	edited	Ash Malyshev	CC BY-SA 4.0	added 11 characters in body
Jan 18 at 0:49	history	edited	Ash Malyshev	CC BY-SA 4.0	added 10 characters in body
Jan 18 at 0:39	history	answered	Ash Malyshev	CC BY-SA 4.0