Timeline for Concentration inequality for max component of a multivariate Gaussian in the general case
Current License: CC BY-SA 4.0
7 events
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| May 29, 2019 at 4:10 | comment | added | ofer zeitouni | Not really, although of course you need to make sure that (say) the maximum of the mean is bounded. | |
| May 28, 2019 at 21:42 | comment | added | ted | Many thanks. So if $f(x) = R^{1/2} x + \mu$, then we have $|f(x) - f(y)| \le \max_i R^{1/2}_{ii} \|(x - \mu) - (y - \mu)\|$ and the means cancel. Btw, why then is Borell's inequality stated for the case of centered Gaussians? (See for example, en.wikipedia.org/wiki/Borell%E2%80%93TIS_inequality) Does something break down for non-centered Gaussians when $T$ is infinite? | |
| May 27, 2019 at 14:51 | comment | added | ofer zeitouni | I didn't mean that the statement is there - just that the proof goes over to the non-centered case. About the rest of your question, I think you are confusing two things: the range of Gaussian variables (which is R), and the index set (which is $T$). In your case, $T=\{1,\ldots,d\}$ and thus $T$ is finite and it is always the case that $E\sup_{t\in T} X_t<\infty$. To summarize - just go over the proof and check that it carries over to your case. | |
| May 27, 2019 at 6:54 | comment | added | ted | Thanks Ofer! Some questions: the first paragraph of section 2 states that the GP is centered, which seems it may not help me. Also, does $\sup_{t\in T} X_t \le \infty$ where $T$ compact imply the sampled Gaussian RVs are assumed to never exceed some max value? If so, this is bad since Gaussian RVs have full support over the real line and therefore have no max value. Lastly, prop 4 does in fact seem helpful since $f$ can rescale and translate the input Gaussian RVs $Y$ s.t. they are no longer centered with have unit variance. Is this the right intuition? Thanks again. | |
| May 26, 2019 at 18:41 | comment | added | ofer zeitouni | The standard proof of Borell's inequality works for this case as well. For example, see page 13 of wisdom.weizmann.ac.il/~zeitouni/notesGauss.pdf | |
| May 25, 2019 at 8:05 | history | edited | ted | CC BY-SA 4.0 |
added 129 characters in body; edited tags; edited title
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| May 23, 2019 at 19:32 | history | asked | ted | CC BY-SA 4.0 |