Timeline for Inverse marginal property of a collection of $\sigma$-algebras
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Apr 27, 2020 at 19:22 | comment | added | Ivan Feshchenko | If $\mathcal{F}_1\subset...\subset\mathcal{F}_n$, then the collection $\mathcal{F}_1,...,\mathcal{F}_n$ does not possess the IMP (if the $\sigma$-algebras are not trivial). | |
| Apr 27, 2020 at 19:06 | comment | added | Ivan Feshchenko | For examples of collections of $\sigma$-algebras that possess the IMP see Section 5 of my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space" (it is available on ArXiv). | |
| Apr 27, 2020 at 19:02 | comment | added | Ivan Feshchenko | If $\mathcal{F_1}\subset\mathcal{F}_2\subset...\subset\mathcal{F}_n$, then the collection $\mathcal{F}_1,...,\mathcal{F}_n$ does not possess the IMP (if the probability space is not trivial). In fact, if $\xi_1,...,\xi_n$ are random variables that satisfy conditions (1),(2),(3) in the Question, then the needed $\xi$ exists if and only if $\xi_1,...,\xi_n$ is a martingale with respect to $\{\mathcal{F}_1,...,\mathcal{F}_n\}$. If this is the case, then one can take $\xi=\xi_n$. | |
| Apr 27, 2020 at 18:57 | comment | added | Ivan Feshchenko | In the definition of the IMP the needed $\xi$ must exist for arbitrary random variables $\xi_1,...,\xi_n$ that satisfy (1), (2), (3) in the Question. | |
| Apr 27, 2020 at 15:01 | history | answered | zab | CC BY-SA 4.0 |