Timeline for Defining density of a random function using Radon-Nikodym Theorem
Current License: CC BY-SA 3.0
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| Dec 5, 2014 at 15:25 | comment | added | SBF | Density is just one representation of a probability measure (through another measure) and it is useful as a tool, when it comes up naturally, not as a goal: by itself it does not tell anything new about probabilities. Now, in the space you've described a natural measure would be the Wiener measure as @Jochen mentioned, but if I'm not mistaken your $P_X$ is singular w.r.t. the Wiener measure. The question is what were you going to do after you'd get a density - if you specify that, I guess the conclusion would be that you don't really need density. | |
| Dec 1, 2014 at 18:46 | comment | added | Janak | I understand the space of such parametric curves is no longer a linear space. But I think it could be a valid question to ask whether the density of the random curve are exists or not. | |
| Dec 1, 2014 at 18:32 | comment | added | Janak | The context of my question is: I have a set of parametric curves $\beta(t)=(x(t),y(t))', t\in [0,1]$ (for example $x(t)=sin t$ and $y(t)=cos t$). I consider these curves are random sample from population of parametric curves and I try to understand whether I can define probability density function of this random curves. | |
| Dec 1, 2014 at 16:04 | comment | added | Jochen Wengenroth | To stimulate this thread you should try to say more precisely what you are looking for. As stated, your question does not make much sense (you could take $\mu=P_X$ and density $1$). For $E=C([0,\infty))$ the Wiener measure (distribution of Brownian motion) is certainly an interesting measure on $E$, but without more context it is not clear whether this is an interesting case for you. | |
| Nov 28, 2014 at 10:37 | comment | added | Janak | @TheBridge: Yes, $P_X$ is push forward measure. | |
| Nov 28, 2014 at 10:33 | comment | added | The Bridge | @ janak : I think that what you want to say is that $P_X$ is the push forward measure of $P$ by $X$. Best regards | |
| Nov 28, 2014 at 8:50 | history | edited | Janak | CC BY-SA 3.0 |
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| Nov 28, 2014 at 8:20 | history | edited | Janak | CC BY-SA 3.0 |
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| Nov 28, 2014 at 8:11 | history | edited | Janak | CC BY-SA 3.0 |
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| Nov 28, 2014 at 7:01 | history | edited | Janak | CC BY-SA 3.0 |
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| Nov 28, 2014 at 6:48 | history | edited | Janak |
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| Nov 28, 2014 at 6:39 | review | First posts | |||
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| Nov 28, 2014 at 6:39 | history | asked | Janak | CC BY-SA 3.0 |