Timeline for Convergence of characteristic functions vs. weak convergence of measures and the Ito-Nisio theorem
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 16, 2020 at 18:07 | comment | added | Abdelmalek Abdesselam | Well, my reference is a set of handwritten notes I wrote for a grad course I taught. As far as publicly available references, first there is the original one which is the article that came out of the thesis by Xavier Fernique aif.centre-mersenne.org/item/AIF_1967__17_1_1_0 but it is in French. Another useful reference is the pedagogical article by Bierme et al digitalcommons.lsu.edu/cgi/… This is probably the best place to start. | |
| Dec 16, 2020 at 7:24 | comment | added | 0xbadf00d | @AbdelmalekAbdesselam Thank you for your comment. Do you have a reference for that? | |
| Dec 15, 2020 at 14:20 | comment | added | Abdelmalek Abdesselam | No the implication does not fail because of "infinite dimension". It does because of the use of Banach spaces. If instead $E$ is a space of distributions like $\mathscr{D}'$ or $\mathscr{S}'$ then Levy's Continuity Theorem holds just like in finite dimensions. In other words tightness follows from pointwise convergence to a function that is continuous at zero. | |
| Dec 15, 2020 at 13:50 | comment | added | zhoraster | Yes, the implication fails. | |
| Dec 15, 2020 at 8:59 | comment | added | 0xbadf00d | @zhoraster Yes, I see, the tightness is the crucial property. In finite dimensions, one is using that if the characteristic functions are equicontinuous at $0$ (which is equivalent to require that they converge to a function which is continuous at $0$), then the corresponding family of measures is tight. I guess this implication fails to hold in infinite dimensions? | |
| Dec 14, 2020 at 8:40 | comment | added | zhoraster | How would you deduce tightness from the convergence of characteristic functions? | |
| Dec 14, 2020 at 5:48 | history | asked | 0xbadf00d | CC BY-SA 4.0 |