Timeline for What happens to the Gaussian volume of a Borel set when it is translated?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 21, 2021 at 10:03 | vote | accept | dohmatob | ||
| Apr 21, 2021 at 10:07 | |||||
| Mar 25, 2019 at 13:16 | comment | added | dohmatob | I've just posted an answer below, based on Neymann-Pearson theory (1933). Lemme know what you think. Thanks. | |
| Mar 21, 2019 at 21:28 | comment | added | Iosif Pinelis | I think, again in view of the highlighted center-of-Gaussian-mass interpretation, the Gaussian content of $A$ is only relevant as a kind of constant factor. I also don't think that this problem has much to do with isoperimetric problems. | |
| Mar 21, 2019 at 20:51 | comment | added | dohmatob | Thanks for the clarification. What if $c$ is a "small" vector, compared to the volume content of $A$, $\|c\| \le \Phi^{-1}(\gamma_n(A))$, with $\gamma_n(A) \ge 1/2$ ? | |
| Mar 21, 2019 at 20:47 | comment | added | Iosif Pinelis | In view of this center-of-Gaussian-mass interpretation, much will likely depend on the choice of $A$ (and $c$, too). Even some of the considerations in my previous comment for the simplest case of $n=1$ and a convex $A$ are not quite trivial. So, I'd be greatly (and pleasantly) surprised if bounds for general $A$ are obtained. | |
| Mar 21, 2019 at 20:20 | comment | added | dohmatob | Can the integral / differential computations in your answer be used to provide bounds (of any sort) on $\gamma_n(c+A)$ as asked in the question, or does your answer say the question is intricately complicated ? Thanks in advance. | |
| Mar 21, 2019 at 20:19 | comment | added | dohmatob | Oops, indeed my computations don't work. Thanks | |
| Mar 21, 2019 at 20:06 | comment | added | Iosif Pinelis | Thank you for your comment. However, in general the equality $E(Z|Z\in c+A)=E(Z-c|Z\in A)$ will not hold. E.g., let $n=1$ and $A=[-1,1]$. Then for large $c>0$ we have $E(Z|Z\in c+A)\sim c$ but $E(Z-c|Z\in A)=-c$. Even the equality $E(Z|Z\in c+A)=E(Z+c|Z\in A)$ will not hold here: for these $n$ and $A$, we have $E(Z|Z\in c+A)<c=E(Z+c|Z\in A)$ for all real $c>0$. | |
| Mar 21, 2019 at 17:52 | comment | added | dohmatob | Great answer, thanks. Possible simplification: Let $\mu_A \in \mathbb R^n$ be the Gaussian barycenter of $A$. Then $E(Z|Z \in c + A) = E(Z-c|Z \in A) = E(Z|Z \in A)-c = \mu_A-c$. | |
| Mar 21, 2019 at 16:35 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |