Timeline for Existence of preferred direction for a random vector with arbitrary distribution on sphere, under a condition on its covariance matrix
Current License: CC BY-SA 4.0
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| Feb 8, 2022 at 13:34 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Feb 8, 2022 at 13:27 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Feb 8, 2022 at 13:21 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Feb 8, 2022 at 12:59 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Feb 8, 2022 at 8:26 | vote | accept | dohmatob | ||
| Feb 8, 2022 at 8:21 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Feb 8, 2022 at 2:29 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Feb 8, 2022 at 2:08 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Feb 8, 2022 at 1:53 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Feb 8, 2022 at 1:40 | comment | added | dohmatob | Sorry for the confusion. $S_{d-1}$ should be $S_{n-1}$. The stray $s_1$ should be the largest eigenvalue of $\Sigma$ (somehow my writeup go scrambled). By arbitrary distribution I meant to say we don't assume the distribution is uniform. Fixed. | |
| Feb 8, 2022 at 1:27 | comment | added | Iosif Pinelis | Previous comment continued: Then you also have an assumption involving some undefined $s_1$, which (I guessed) also probably contradicts the "an arbitrary distribution on the unit-sphere" assumption. | |
| Feb 8, 2022 at 1:26 | comment | added | Iosif Pinelis | @dohmatob : Your post is indeed quite confusing, and I really had to guess, several times, what you mean. Your begin with "Let $X$ be random vector with an arbitrary distribution on the unit-sphere $S_{d-1}$ in $\mathbb R^n$." (So, I guessed, for you $d$ is $n$.) So, my answer corresponds to this. Further, you say "For example, if $X$ has standard gaussian iid components, then (1) holds." But the latter assumption contradicts the above one, about "an arbitrary distribution on the unit-sphere"; so, I thought it may be dismissed. | |
| Feb 8, 2022 at 1:12 | comment | added | dohmatob | Thanks for the input. But this is an answer to a question I never asked :). Indeed, I'm aware that (1) should is impossible in general (i.e without further conditions on $X$). I'm only interested in establishing inequalities of the form (1) under the assumption explicitly started in my question. For your example, one has $s_1 = 1/n$, which precisely violates the assumption in my question. No ? | |
| Feb 8, 2022 at 0:33 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |