Timeline for Anti-concentration of inner product of a uniform unit norm vector with another vector
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 1, 2021 at 18:34 | comment | added | jlewk | Yes; to overcome dependence I had in mind to work instead with $b_0(1\pm\epsilon)$ in the event $|\sqrt{\chi^2_d/d}-1|\le \epsilon$ that would not affect the probability of the event by more than $2e^{-\epsilon^2 d/2}$. | |
| Aug 1, 2021 at 18:28 | answer | added | Yuval Peres | timeline score: 1 | |
| Aug 1, 2021 at 17:28 | comment | added | Yuval Peres | @jlewk : Note that $Z$ and $\chi_d^2$ are dependent. | |
| Aug 1, 2021 at 10:23 | comment | added | jlewk | WLOG assume $\|a\|=1$ and $b=b_0/\sqrt d$. Write $X=g/\|g\|$ with $g\sim N(0,I_d)$. Then your event has the form $\{Z>b_0 \sqrt{\chi^2_d/d} \}$ with $Z=g^Ta \sim N(0,1)$ and $\chi^2_d$ chi-square with $d$ degrees of freedom. Exponential concentration of the chi-square should give a solution. | |
| Aug 1, 2021 at 9:50 | history | asked | Probabilist | CC BY-SA 4.0 |