Timeline for When do maximum and expectation commute?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Sep 24, 2012 at 0:50 | comment | added | martin | That indeed is a very strong condition :) | |
| Sep 23, 2012 at 20:48 | comment | added | Stanislav | This step requires $\sum a_{kn} t^k E(X^n)$ to be an increasing function of $t$. It may be a very strong condition, but I don't see any other way to exchange $\sup(\cdot)$ and $E(\cdot)$ here. | |
| Sep 23, 2012 at 20:22 | comment | added | martin | Is $\sup_t\sum a_{kn}t^kE(x^n)=\sum a_{kn}\sup_t(t^k)E(x^n)$ a typo? How can you pass $\sup$ into the summation? | |
| Sep 23, 2012 at 19:47 | comment | added | Stanislav | If $X$ keeps its sign, one can also consider the Taylor series of $G(t,x)$: $G(t,x) = \sum_{k,n\ geq 0} a_{kn} t^k x^n$ and analyze the conditions under which $\sup_t E(\sum a_{kn} t^k x^n) = \sup_t \sum a_{kn} t^k E(x^n) = \sum a_{kn} \sup_t(t^k) E(x^n) = E(\sum a_{kn} \sup_t(t^k) x^n) = E( \sup_t( \sum a_{kn} t^k x^n))$. | |
| Sep 23, 2012 at 14:30 | comment | added | martin | Thanks, but I would like necessary and sufficient conditions on $G(t,x)$ (or at least a condition that allows for a more general class of functions $G(t,x)$ if possible). | |
| Sep 23, 2012 at 12:04 | history | edited | Stanislav | CC BY-SA 3.0 |
added 57 characters in body
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| Sep 23, 2012 at 11:56 | history | answered | Stanislav | CC BY-SA 3.0 |