Timeline for Normal distribution by successive approximation?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Jan 17, 2023 at 2:09 | vote | accept | Iosif Pinelis | ||
| Jan 17, 2023 at 0:37 | answer | added | fedja | timeline score: 1 | |
| May 31, 2021 at 16:22 | comment | added | Iosif Pinelis | @AbdelmalekAbdesselam : Thank you for these further ideas. Meanwhile, i have done some simple rewriting of the iteration equation, now almost in a convolution form. | |
| May 31, 2021 at 16:18 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| May 31, 2021 at 16:05 | comment | added | Abdelmalek Abdesselam | ...in the near Gaussian case. Given that the situation is simpler than for the RG, I would expect it would not require as much ingenuity to find a Lyapunov functions that would allow a proof in the global (far from Gaussian) case. BTW a better link than above is mathoverflow.net/questions/182752/… | |
| May 31, 2021 at 16:05 | comment | added | Abdelmalek Abdesselam | I just did a quick computation which gives support to your conjecture. Functions $h_k(r)=r^k$ are eigenfunctions of the linearization (no need for orthogonal polynomials like Hermite etc.) with eigenvalues $c_k=(4/\pi)\times W_k$ in terms of Wallis integrals in the notations of en.wikipedia.org/wiki/Wallis%27_integrals The only expanding/relevant directions are for $k=0,1$, while $k=2$ is neutral/marginal. Finally for $k>2$ , the corresponding directions are contracting. This is exactly the same as in the RG link I mentioned. One can mimick Koralov-Sinai and make this into a proof... | |
| May 31, 2021 at 15:50 | comment | added | Iosif Pinelis | @AbdelmalekAbdesselam : Thank you for your suggestions. I see your point better now, and will have these suggestions in mind. | |
| May 31, 2021 at 15:11 | comment | added | Abdelmalek Abdesselam | Of course the iterations are different, but the methods used for one might work for the other. It would take me some time to work it out, but the renormalization group inspired strategy I would use for your questions is: 1) write $g$ as the Gaussian times $(1+h)$ and write the iteration for the function $h$, 2) write the linearization of the transformation at the fixed point $h=0$, 3) see if you can diagonalize it explicitly using suitable orthogonal polynomials for the integral over $t$. Ultimately, one would need a Lyapunov function like some kind of entropy. Does Stein's method help? | |
| May 31, 2021 at 15:03 | comment | added | Iosif Pinelis | @AbdelmalekAbdesselam : Thank you for this reference. Both settings indeed involve iterations. I think that other setting is a case of the central limit theorem for iid summands and $2^k$ summands, for natural $k$. Here the setting involves a different kind of iterations. But presumably/hopefully in this setting too one has normality in the limit. | |
| May 31, 2021 at 14:46 | comment | added | Abdelmalek Abdesselam | Related mathoverflow.net/questions/191791/… | |
| May 31, 2021 at 13:19 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| May 31, 2021 at 0:54 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| May 31, 2021 at 0:49 | history | asked | Iosif Pinelis | CC BY-SA 4.0 |