Timeline for Extreme Points of a set of distributions with moment and/or support constraint
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 16, 2017 at 16:44 | comment | added | Mateusz Kwaśnicki | I meant $P_1$; it need not be extremal for $P_2$. If $F$ is extremal for $P_3$ and $F=(1-t)F_1+t F_2$, $F_1, F_2 \in P_1$, $t\in(0,1)$, then in fact $F_1, F_2 \in P_3$ (because the support of $F_1$ and $F_2$ is contained in the support of $F$) and therefore $F_1=F_2=F$. Hence, $F$ is extremal for $P_1$. Converse is always true: an extremal point $F$ of a set is also extremal for any subset that contains $F$. | |
| Aug 16, 2017 at 13:06 | comment | added | Boby | In the last part, do you mean that $F \in \mathbb{P}_3$ is extremal for $P_3$ is and only if it is extremal for $P_2$? Or do you mean $P_1$? | |
| Aug 15, 2017 at 21:27 | history | bounty ended | Boby | ||
| Aug 15, 2017 at 21:27 | vote | accept | Boby | ||
| Aug 15, 2017 at 21:00 | comment | added | Mateusz Kwaśnicki | No, I do not have any reference. Dirac measures are the extreme points of the entire set of probability measures. Indeed, if $\delta_x=(1-t)\mu_1+t\mu_2$, then both $\mu_1$ and $\mu_2$ are necessarily supported in $\{x\}$, and so they are equal to $\delta_x$. Conversely, if $\mu$ is a measure with support $A$ that contains at least two points, then $A$ can be written as $A=A_1\cup A_2$ with $t:=\mu(A_2)\in(0,1)$ and so $\mu=(1-t)\mu_1+t\mu_2$ with $\mu_j(E)=\mu(E\cap A_j)/\mu(A_j)$. Thus, $\mu$ is not extremal. This argument is valid also when the support of $\mu$ is restricted to a given set. | |
| Aug 15, 2017 at 0:13 | comment | added | Boby | Do you have a good reference on this subject? Also, never understood by why $\delta_x$ are extreme points in the case when $|x| \le d$. Could you explain this point a little? | |
| Aug 11, 2017 at 22:27 | history | answered | Mateusz Kwaśnicki | CC BY-SA 3.0 |