Timeline for How to show that quadratic mean convergence implies expectation value?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 20, 2020 at 17:57 | answer | added | Thiago Bittencourt | timeline score: 0 | |
| S Jan 29, 2020 at 17:23 | history | bounty ended | an1lam | ||
| S Jan 29, 2020 at 17:23 | history | notice removed | an1lam | ||
| Jan 28, 2020 at 14:03 | vote | accept | an1lam | ||
| Jan 28, 2020 at 5:43 | answer | added | Anon | timeline score: 4 | |
| Jan 22, 2020 at 18:00 | history | tweeted | twitter.com/StackStats/status/1220043606603550720 | ||
| S Jan 22, 2020 at 16:34 | history | bounty started | an1lam | ||
| S Jan 22, 2020 at 16:34 | history | notice added | an1lam | Draw attention | |
| Dec 29, 2019 at 23:31 | comment | added | Sean Roberson | Precisely. I'll do a full solution when I can figure the second part. | |
| Dec 29, 2019 at 22:28 | comment | added | an1lam | Oh I think I see now: you can let the $ Y $ variable in Cauchy-Schwarz just be $ 1 $ and then you get $$ \mathbb{E}(\lvert X_n - b \rvert) \leq \sqrt{\mathbb{E}(X_n - b)^2} $$ which goes to $ 0 $ by assumption. | |
| Dec 26, 2019 at 20:49 | comment | added | an1lam | Maybe I'm just being dense but I don't see how the Cauchy-Schwarz inequality helps with the inequality chain you started. The probability version of Cauchy-Schwarz that I'm familiar with is $ \mathbb{E}(X^2Y^2) \leq \mathbb{E}(X^2) \mathbb{E}(Y^2) $. How does that relate to $ \mathbb{E} \lvert X_n - b \rvert \leq \dots $? | |
| Dec 25, 2019 at 17:06 | comment | added | an1lam | Ah I see. I have to check but I think we can assume the limits exist (as in it's in the problem statement). | |
| Dec 25, 2019 at 5:27 | comment | added | Sean Roberson | Also: your last line only holds if all limits exist and are finite. We don't know if $E(X_n)$ exists! | |
| Dec 25, 2019 at 5:12 | comment | added | Sean Roberson | Just a hint: If $E(X_n) \to b$ in $L^2$, then we can have $\int |X_n - b| \ dP \leq \ldots$ Use Cauchy-Schwarz! | |
| Dec 25, 2019 at 3:40 | review | Close votes | |||
| Dec 25, 2019 at 12:45 | |||||
| Dec 25, 2019 at 3:05 | review | First posts | |||
| Dec 25, 2019 at 3:20 | |||||
| Dec 25, 2019 at 3:00 | history | asked | an1lam | CC BY-SA 4.0 |