Timeline for extensions to Bernstein's inequality I could use for bounding probability of a union?
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| Oct 8, 2010 at 9:06 | comment | added | Hugh J | The assumptions depend on the applications you have in mind. If you really have no applications in mind and are just trying to state an abstract result, then the best assumption is probably : $P(\cup A_i)$ is small enough". Note that if $N\epsilon$ is large and if you want the probability of the union to be small you require strong assumptions on the dependence. It would be nice to have at least non trivial examples (other that all the $A_i$ are equal for example) to make sure that your result is non trivial. | |
| Oct 5, 2010 at 12:24 | comment | added | Mark Meckes | Good point, Bill, and more generally the dependency structure among Bernoullis can't be terribly complicated, so in such a case the Bonferroni inequalities are particularly likely to be useful. But it's impossible to be sure without clarification. I think, though, that it's worth waiting at least 24 hours for clarification before closing, just on general principle. (It's entirely academic to me, since I don't have sufficient rep to vote for closing.) | |
| Oct 5, 2010 at 1:02 | comment | added | Bill Johnson | Sure, Mark, but a martingale difference sequence of Bernoulli's is independent. It is hard to make sense of this question. Since the OP hasn't clarified, I vote to close. | |
| Oct 4, 2010 at 15:14 | comment | added | Mark Meckes | Also, Bernstein-like inequalities hold under martingale dependence assumptions; see for example en.wikipedia.org/wiki/Azuma%27s_inequality It's hard to say what will be useful here without some more detail about what kind of dependence assumptions are acceptable to you. | |
| Oct 4, 2010 at 15:11 | comment | added | Mark Meckes | Depending on the assumptions you're willing to make, one useful refinement of the union bound is the Bonferroni inequalities: en.wikipedia.org/wiki/… | |
| Oct 4, 2010 at 15:01 | history | asked | claude | CC BY-SA 2.5 |