Timeline for Bochner-Minlos for moment-generating functions?
Current License: CC BY-SA 4.0
11 events
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| Mar 16, 2021 at 6:55 | vote | accept | iolo | ||
| Mar 15, 2021 at 18:29 | history | edited | Abdelmalek Abdesselam |
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| Feb 18, 2021 at 12:32 | comment | added | Abdelmalek Abdesselam | ...can give you pointwise convergence of characteristic functions far away from the origin. However, one still needs other arguments to prove the continuity at the origin of the full infinite-dimensional limit characteristic function. I will edit my answer with a specific worked out example that I lectured about in a graduate course I taught a while ago. | |
| Feb 18, 2021 at 12:29 | comment | added | Abdelmalek Abdesselam | ...something to work out and turn into a quick research paper in their first two months of PhD work. Now even if the ideal theorem is not yet available, there are some hacks one can already do in specific situations (that's why I asked for specific details). For example one can replace $i\varphi(f)$ in the exponential featuring in the characteristic function by $z\varphi(f)$ where $z$ is a small complex number. For fixed $f$, this is about uniform convergence of holomorphic functions of one complex variable. Not scary at all. This approach, in the spirit of the Cramer-Wold device,... | |
| Feb 18, 2021 at 12:20 | comment | added | Abdelmalek Abdesselam | Thanks for the infos, although you didn't say which references you were following. I an ideal world, there would be an article which proves the theorem you need. Here is a metatheorem: if your moment generating functions are well defined and analytic in a complex neighborhood of the origin and if you convergence is reasonably uniform in that neighborhood, then you have weak convergence of probability measures. I don't think you will find this in the literature which, in this area, is a giant mess. I'm confident this is true, meaning, if I had a new PhD student I could assign this as... | |
| Feb 18, 2021 at 8:06 | comment | added | iolo | So many questions :D Well, the measures indeed live on the space of tempered distributions as in your answer. They correspond to QTFs in the sense of Osterwalder-Schrader but are regularized - but this is more of a goal than an à priori feature. I obtain the moment-generating functions rather indirectly and I only really have control over the Fenchel conjugate of the limiting object. To be honest, I would not know how to perform an analytic continuation in this setting at all which is of course another reason to ask this question - begging there could exist a simple answer. | |
| Feb 17, 2021 at 15:45 | history | edited | Abdelmalek Abdesselam |
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| Feb 17, 2021 at 0:35 | answer | added | Abdelmalek Abdesselam | timeline score: 4 | |
| Feb 16, 2021 at 18:11 | history | edited | Abdelmalek Abdesselam |
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| Feb 16, 2021 at 17:47 | comment | added | Abdelmalek Abdesselam | A good question, but one would need a lot more specific details in order to be able to help you. What are these measures, and on what space? What is the extent of your control of the moment generating function? How come you can't do a little bit of analytic continuation and get a hold of the characteristic function instead? What do you know already, and what references did you look at? etc. etc. | |
| Feb 16, 2021 at 11:12 | history | asked | iolo | CC BY-SA 4.0 |