Timeline for About covariance operators for probability distributions on a function space
Current License: CC BY-SA 3.0
10 events
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| Aug 11, 2016 at 19:27 | comment | added | Iosif Pinelis | Any (say) continuous square-integrable random process $X=(X(t))_{t\in I}$ (such as the Wiener process) on a closed bounded interval $I$ can indeed be viewed as a random function in $L^2(I)$. Assume, for simplicity of writing, that $E X=0$ (as is the case with the Wiener process). Then the covariance operator $R$ of $X$ is the integral transform with kernel function $I^2\ni(t,s)\mapsto K(t,s):=EX(t)\overline{X(s)}$. That is, $(Rx)(t)=E\langle x,X\rangle X(t)=EX(t)\int_I\overline{X(s)}x(s)\,ds=\int_I K(t,s)x(s)\,ds$ for all $x\in L^2(I)$ and $t\in I$. | |
| Aug 11, 2016 at 18:11 | vote | accept | gradstudent | ||
| Aug 11, 2016 at 18:11 | comment | added | gradstudent | Is there any analogous "Wiener distribution" and corresponding such spectral decomposition for the covariance matrix on a more general space of $\mathbb{R}^n \rightarrow \mathbb{R}^m$ functions? | |
| Aug 11, 2016 at 18:09 | comment | added | gradstudent | Thanks! So this covariance operator is of the ``Wiener distribution" on the space of $\mathbb{R} \rightarrow \mathbb{R}$, right? Here the "stochastic process" is just an intermediary and not really the important point - am I right? And the $W_t$ whose spectral decomposition is given here, en.wikipedia.org/wiki/Wiener_process#Wiener_representation is really the value of a randomly sampled function using the the Wiener distribution. Right? | |
| Aug 7, 2016 at 4:34 | comment | added | Iosif Pinelis | I have added a paragraph on the Karhunen--Loève decomposition, including the specific examples of the Brownian motion and the Brownian bridge. | |
| Aug 7, 2016 at 4:32 | history | edited | Iosif Pinelis | CC BY-SA 3.0 |
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| Aug 6, 2016 at 22:12 | comment | added | gradstudent | Thanks for this very helpful reference! Would you know if there are special cases of a distribution over an infinite dimensional function space where one explicitly knows these eigenfunctions of the covariance operator? | |
| Aug 5, 2016 at 19:52 | history | edited | Iosif Pinelis | CC BY-SA 3.0 |
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| Aug 5, 2016 at 17:44 | history | edited | Iosif Pinelis | CC BY-SA 3.0 |
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| Aug 5, 2016 at 16:50 | history | answered | Iosif Pinelis | CC BY-SA 3.0 |