Timeline for Mean square derivatives and modifications
Current License: CC BY-SA 4.0
7 events
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| Jan 25, 2021 at 18:20 | comment | added | Iosif Pinelis | @123456 : The inequality in your last comment is correct, but only because the latter expectation is $0$. It is better to use here Minkowski's inequality $\|U+V\|_2\le\|U\|_2+\|V\|_2$ or, alternatively, the inequality $(U+V)^2\le2U^2+2V^2$.Using the latter inequality, you will need the extra factor $2$ on the right-hand side of your inequality. | |
| Jan 25, 2021 at 17:49 | comment | added | 123 456 | Might be an obvious question but is the following correct: if $Z(.)$ is a modification of $X'(.)$, then $Z(.)$ is also a mean square derivative of $X(.)$, because for each $t$, $$\mathbb{E}\left[\left(\frac{X(t+h)-X(t)}{h}- Z(t)\right)^2\right] \leq \mathbb{E}\left[\left(\frac{X(t+h)-X(t)}{h}-X'(t)\right)^2\right]+ \mathbb{E}\left[\left(X'(t)-Z(t)\right)^2\right]$$ the first term tends to 0 as $h \to 0$ by definition, and the second term is 0 because $X'(t)$ and $Z(t)$ are equal almost surely by the fact that $Z(.)$ is a modification of $X'(.)$? | |
| Jan 24, 2021 at 19:12 | vote | accept | 123 456 | ||
| Jan 24, 2021 at 19:12 | |||||
| Jan 24, 2021 at 15:46 | comment | added | Iosif Pinelis | @123456 : I have elaborated on this. | |
| Jan 24, 2021 at 15:45 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 263 characters in body
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| Jan 24, 2021 at 5:42 | comment | added | 123 456 | Thanks. Could you elaborate a bit more on why modifications are preserved in the limit? In particular why $Y'(.)$ is a modification of $X'(.)$ because they are the limits of two sequences that are modifications of each other. | |
| Jan 24, 2021 at 2:53 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |