Timeline for Projection of log-concave distribution on unit sphere surface
Current License: CC BY-SA 4.0
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| Jun 6, 2023 at 18:07 | comment | added | entechnic | I'm mainly interested in finding a general upper bound for the high-dimensional case, as function of the dimension $d$ (not necessarily a very tight one). The motivation comes from the multivariate standard Gaussian distribution, whose projection on the unit sphere is uniform. I would like to see what happens in the more general case of log-concave distributions and if there is any relation with the Gaussian case (for instance, in terms of the ratio between the requested upper bound and the uniform value) | |
| Jun 6, 2023 at 15:10 | comment | added | Dan | Can you clarify what kind of upper bound you're interested in, or your motivation? This might help lead to a useful answer. I am not sure if the high-dimensional case is of interest to you, but if it is, then the lower-dimensional marginals of $X/\|X\|_2$ are approximately Gaussian, and this can be made quantitative using Klartag's central limit theorem combined with thin shell estimates, Theorems 1.1 and 1.4 here: link.springer.com/article/10.1007/s00222-006-0028-8 | |
| May 31, 2023 at 23:35 | history | edited | entechnic | CC BY-SA 4.0 |
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| S May 30, 2023 at 13:48 | review | First questions | |||
| May 30, 2023 at 14:30 | |||||
| S May 30, 2023 at 13:48 | history | asked | entechnic | CC BY-SA 4.0 |