Uncorrelatedness of an Arbitrary Collection of Random Variables
$begingroup$
I have a probability space $(Omega,mathcal F, P)$ and a rv $ X:(Omega,mathcal F)rightarrow(Sigma,mathcal B)$, where $mathcal B$ is the Borel-$sigma$-algebra generated by a complete inner product defined on $Sigma$.
Lang shows that this is enough to define the Lebesgue integral and thus also the expectation $mathbb E[X],{:=} int X mathsf dP$, provided that $mathbb E[Vert XVert]<infty$.
Now let $(X_i)_{iin I}$ be a collection of rv in the above setting. I wonder how do we define uncorrelatedness for that collection? Since $Sigma$ may not be the Euclidean space, there is no notion of a product and we can't get a result like $mathbb Eleft[prod_{iin I}X_iright] = prod_{iin I}mathbb E[X_i]$ $iff$ $(X_i)_{iin I}$ is uncorrelated.
However, there exists a covariance operator (there are a number of questions and answers to that here on Math.SE) defined by
$$mathrm{Cov}[X_i,X_j](h):{:=}int langle X_i, hrangle X_j$$
(assuming that all the random variables are centered, i.e. have mean zero). But this definition does not allow an arbitrary number of elements since an inner product (and the scalar product) is only defined for two input values. So how can I check an arbitrary collection? Or are these thoughts for some reason are overlooking something and there is not even a notion of uncorrelatedness for more abstract spaces?
Regarding another concept, independence of random variables, there is a nice generalization: One can easily define $(X_i)_{iin I}$ to be independent if and only if the joint measure equals the product measure.
functional-analysis probability-theory hilbert-spaces
asked Dec 18 '18 at 0:23
Syd AmerikanerSyd Amerikaner
133213
$endgroup$
add a comment |
$begingroup$
I have a probability space $(Omega,mathcal F, P)$ and a rv $ X:(Omega,mathcal F)rightarrow(Sigma,mathcal B)$, where $mathcal B$ is the Borel-$sigma$-algebra generated by a complete inner product defined on $Sigma$.
Lang shows that this is enough to define the Lebesgue integral and thus also the expectation $mathbb E[X],{:=} int X mathsf dP$, provided that $mathbb E[Vert XVert]<infty$.
Now let $(X_i)_{iin I}$ be a collection of rv in the above setting. I wonder how do we define uncorrelatedness for that collection? Since $Sigma$ may not be the Euclidean space, there is no notion of a product and we can't get a result like $mathbb Eleft[prod_{iin I}X_iright] = prod_{iin I}mathbb E[X_i]$ $iff$ $(X_i)_{iin I}$ is uncorrelated.
However, there exists a covariance operator (there are a number of questions and answers to that here on Math.SE) defined by
$$mathrm{Cov}[X_i,X_j](h):{:=}int langle X_i, hrangle X_j$$
(assuming that all the random variables are centered, i.e. have mean zero). But this definition does not allow an arbitrary number of elements since an inner product (and the scalar product) is only defined for two input values. So how can I check an arbitrary collection? Or are these thoughts for some reason are overlooking something and there is not even a notion of uncorrelatedness for more abstract spaces?
Regarding another concept, independence of random variables, there is a nice generalization: One can easily define $(X_i)_{iin I}$ to be independent if and only if the joint measure equals the product measure.
functional-analysis probability-theory hilbert-spaces
asked Dec 18 '18 at 0:23
Syd AmerikanerSyd Amerikaner
133213
$endgroup$
$begingroup$
I would say that a family $mathcal F$ is pairwise (perhaps pairwisely?) uncorrelated if $mathbb E(langle X,Yrangle)=0$ for all distinct $X,Yin mathcal F$.
$endgroup$
– Jochen
Dec 18 '18 at 8:49
add a comment |
$begingroup$
I have a probability space $(Omega,mathcal F, P)$ and a rv $ X:(Omega,mathcal F)rightarrow(Sigma,mathcal B)$, where $mathcal B$ is the Borel-$sigma$-algebra generated by a complete inner product defined on $Sigma$.
Lang shows that this is enough to define the Lebesgue integral and thus also the expectation $mathbb E[X],{:=} int X mathsf dP$, provided that $mathbb E[Vert XVert]<infty$.
Now let $(X_i)_{iin I}$ be a collection of rv in the above setting. I wonder how do we define uncorrelatedness for that collection? Since $Sigma$ may not be the Euclidean space, there is no notion of a product and we can't get a result like $mathbb Eleft[prod_{iin I}X_iright] = prod_{iin I}mathbb E[X_i]$ $iff$ $(X_i)_{iin I}$ is uncorrelated.
However, there exists a covariance operator (there are a number of questions and answers to that here on Math.SE) defined by
$$mathrm{Cov}[X_i,X_j](h):{:=}int langle X_i, hrangle X_j$$
(assuming that all the random variables are centered, i.e. have mean zero). But this definition does not allow an arbitrary number of elements since an inner product (and the scalar product) is only defined for two input values. So how can I check an arbitrary collection? Or are these thoughts for some reason are overlooking something and there is not even a notion of uncorrelatedness for more abstract spaces?
Regarding another concept, independence of random variables, there is a nice generalization: One can easily define $(X_i)_{iin I}$ to be independent if and only if the joint measure equals the product measure.
functional-analysis probability-theory hilbert-spaces
asked Dec 18 '18 at 0:23
Syd AmerikanerSyd Amerikaner
133213
$endgroup$
I have a probability space $(Omega,mathcal F, P)$ and a rv $ X:(Omega,mathcal F)rightarrow(Sigma,mathcal B)$, where $mathcal B$ is the Borel-$sigma$-algebra generated by a complete inner product defined on $Sigma$.
Lang shows that this is enough to define the Lebesgue integral and thus also the expectation $mathbb E[X],{:=} int X mathsf dP$, provided that $mathbb E[Vert XVert]<infty$.
Now let $(X_i)_{iin I}$ be a collection of rv in the above setting. I wonder how do we define uncorrelatedness for that collection? Since $Sigma$ may not be the Euclidean space, there is no notion of a product and we can't get a result like $mathbb Eleft[prod_{iin I}X_iright] = prod_{iin I}mathbb E[X_i]$ $iff$ $(X_i)_{iin I}$ is uncorrelated.
However, there exists a covariance operator (there are a number of questions and answers to that here on Math.SE) defined by
$$mathrm{Cov}[X_i,X_j](h):{:=}int langle X_i, hrangle X_j$$
(assuming that all the random variables are centered, i.e. have mean zero). But this definition does not allow an arbitrary number of elements since an inner product (and the scalar product) is only defined for two input values. So how can I check an arbitrary collection? Or are these thoughts for some reason are overlooking something and there is not even a notion of uncorrelatedness for more abstract spaces?
Regarding another concept, independence of random variables, there is a nice generalization: One can easily define $(X_i)_{iin I}$ to be independent if and only if the joint measure equals the product measure.
functional-analysis probability-theory hilbert-spaces
functional-analysis probability-theory hilbert-spaces
asked Dec 18 '18 at 0:23
Syd AmerikanerSyd Amerikaner
133213
asked Dec 18 '18 at 0:23
Syd AmerikanerSyd Amerikaner
133213
asked Dec 18 '18 at 0:23
Syd AmerikanerSyd Amerikaner
133213
asked Dec 18 '18 at 0:23
Syd AmerikanerSyd Amerikaner
133213
asked Dec 18 '18 at 0:23
Syd AmerikanerSyd Amerikaner
133213
133213
$begingroup$
I would say that a family $mathcal F$ is pairwise (perhaps pairwisely?) uncorrelated if $mathbb E(langle X,Yrangle)=0$ for all distinct $X,Yin mathcal F$.
$endgroup$
– Jochen
Dec 18 '18 at 8:49
add a comment |
$begingroup$
I would say that a family $mathcal F$ is pairwise (perhaps pairwisely?) uncorrelated if $mathbb E(langle X,Yrangle)=0$ for all distinct $X,Yin mathcal F$.
$endgroup$
– Jochen
Dec 18 '18 at 8:49
$begingroup$
I would say that a family $mathcal F$ is pairwise (perhaps pairwisely?) uncorrelated if $mathbb E(langle X,Yrangle)=0$ for all distinct $X,Yin mathcal F$.
$endgroup$
– Jochen
Dec 18 '18 at 8:49
$begingroup$
I would say that a family $mathcal F$ is pairwise (perhaps pairwisely?) uncorrelated if $mathbb E(langle X,Yrangle)=0$ for all distinct $X,Yin mathcal F$.
$endgroup$
– Jochen
Dec 18 '18 at 8:49
add a comment |
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$begingroup$
I would say that a family $mathcal F$ is pairwise (perhaps pairwisely?) uncorrelated if $mathbb E(langle X,Yrangle)=0$ for all distinct $X,Yin mathcal F$.
$endgroup$
– Jochen
Dec 18 '18 at 8:49