Timeline for Can we show that the characteristic function of an infinitely divisible probability measure has no zeros
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jun 30, 2021 at 8:49 | vote | accept | 0xbadf00d | ||
| Jun 27, 2021 at 8:06 | history | bounty ended | CommunityBot | ||
| Jun 25, 2021 at 22:56 | comment | added | Iosif Pinelis | [Previous comment continued:] Anyhow, by the virtue of the above answer, your "actual goal", "to deduce that an infinitely divisible probabilty measures $\mu$ on $\mathcal B(E)$ satisfies $\varphi_\mu(x')\ne0$ for all $x'\in E'$", is now attained. It is also explained why the other approaches will not work. | |
| Jun 25, 2021 at 22:56 | comment | added | Iosif Pinelis | @0xbadf00d : As for the Ito-Nisio theorem, it is essentially about the convergence in distribution of a series of independent symmetric random vectors to a random vector -- rather than about a triangular array of iid (in each row of the array) independent not necessarily symmetric random vectors, as in your case. So, I do not see any possible relevance of the Ito-Nisio theorem to your question. | |
| Jun 24, 2021 at 17:33 | comment | added | Iosif Pinelis | @0xbadf00d : I have added a detail on how to get the desired no-zeroes conclusion using only an intermediate result in a proof of the Lévy-Khintchine formula. | |
| Jun 24, 2021 at 17:29 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 24, 2021 at 15:52 | comment | added | 0xbadf00d | Regarding 2: How do you prove the Lévy-Khintchine formula without already knowing that the characteristic function of an infinite divisible measures has no zeros? It is actually my intent to prove the Lévy-Khintchine formula in the setting of the question and the problem described in my post is one problem which I've encountered in generalization the usual proof. Do you've got another idea how we can see the desired claim without the Lévy-Khintchine formula? Don't you think the Ito-Nisio theorem could be applied, for example? | |
| Jun 24, 2021 at 15:46 | comment | added | 0xbadf00d | Thank you for your answer. Regarding 1: The limit function in question, i.e. $\varphi$ defined in $(4)$, is continuous at the origin. The problem with Lévy's continuity theorem is that it only holds for measures on $\mathcal B(\mathbb R^d$), $d\in\mathbb N$. | |
| Jun 22, 2021 at 19:42 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 22, 2021 at 17:22 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 22, 2021 at 15:37 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |