Timeline for Estimating probability that a large sum of i.i.d variables is positive
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 8, 2019 at 20:07 | comment | added | Iosif Pinelis | It's nice to hear that the conjecture about the logarithmic asymptotics is now confirmed. | |
| May 8, 2019 at 8:49 | comment | added | Bogdan | Of course, it would be even nicer to have the EXACT asymptotic, but, unless I am mistaken, some of the conditions before Theorem 2.1 not only difficult to verify - they just does not hold. It looks like Theorem 2.1 corresponds to one of the "main cases" in [26], while my case is exactly the "intermediate" one covered only by [26, Theorem 5]. | |
| May 8, 2019 at 8:40 | comment | added | Bogdan | My question was about upper and lower bounds, and, inspired by [26, Theorem 5], I was able to rigorously prove some bounds in my case, which, in particular, imply the crude logarithmic asymptotic $\log f(n) \sim c\sqrt{n}$. This is already sufficient for publication. | |
| May 7, 2019 at 17:20 | comment | added | Iosif Pinelis | Previous comment continued: So, I think the best (if not the only) way to proceed here is indeed to go through lots of pretty hard calculations required by the mentioned result by S. V. Nagaev. | |
| May 7, 2019 at 17:18 | comment | added | Iosif Pinelis | @Bogdan : I am glad this was of help. However, I am afraid that the less general result [26, Theorem 5] in the paper by A. V. Nagaev referred to in the paper by his brother S. V. Nagaev will not suffice in your case, because A. V. Nagaev assumes a rather narrow condition on the density -- in his formula (2), which I think the density of your $Z$ will not quite satisfy. A. V. Nagaev does have a footnote statement about this on the first page of his paper, but that statement is not proved and seems too optimistic to me, especially in light of S. V. Nagaev's result. | |
| May 7, 2019 at 16:28 | vote | accept | Bogdan | ||
| May 7, 2019 at 16:27 | comment | added | Bogdan | Thank you! The conditions in Theorem 2.1 are indeed complicated and difficult to verify in my case, but the paper has a reference to another paper (reference [26]) in which Theorem 5 looks like exactly what is needed! The only problem is that $\sigma^2$ in [26, Theorem 5] is undefined, but this already looks like a minor issue. | |
| May 7, 2019 at 1:07 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |