Mixing
Is there a “practical” Hilbert space of stochastic processes?
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It is well-known that one can construct a Hilbert space of zero mean, finite variance random variables such that every random variable of this type $X$ can be represented as a linear combination of a sequence of random variables $F_i$ with respect to probability:
$$X = sum_{i=1}^infty frac{text{cov}(X,F_i)}{text{var}(F_i)}F_i$$
Is there any such meaningful Hilbert space construction for stochastic processes $X(t)$?
probability-theory statistics stochastic-processes hilbert-spaces
edited Jan 31 at 11:37
YuiTo Cheng
2,0392636
asked Jan 23 at 5:29
aghostinthefiguresaghostinthefigures
1,2641217
$endgroup$
add a comment |
$begingroup$
It is well-known that one can construct a Hilbert space of zero mean, finite variance random variables such that every random variable of this type $X$ can be represented as a linear combination of a sequence of random variables $F_i$ with respect to probability:
$$X = sum_{i=1}^infty frac{text{cov}(X,F_i)}{text{var}(F_i)}F_i$$
Is there any such meaningful Hilbert space construction for stochastic processes $X(t)$?
probability-theory statistics stochastic-processes hilbert-spaces
edited Jan 31 at 11:37
YuiTo Cheng
2,0392636
asked Jan 23 at 5:29
aghostinthefiguresaghostinthefigures
1,2641217
$endgroup$
$begingroup$
Is this what you have in mind? en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem
$endgroup$
– Mike Hawk
Jan 28 at 16:25
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@MikeHawk That’s essentially it—although I expected deterministic coefficients on a sequence of stochastic processes instead of stochastic coefficients on a sequence of deterministic processes, I’ll take what I can get; feel free to put it down as an answer and I’ll give you the bounty!
$endgroup$
– aghostinthefigures
Jan 28 at 16:40
add a comment |
$begingroup$
It is well-known that one can construct a Hilbert space of zero mean, finite variance random variables such that every random variable of this type $X$ can be represented as a linear combination of a sequence of random variables $F_i$ with respect to probability:
$$X = sum_{i=1}^infty frac{text{cov}(X,F_i)}{text{var}(F_i)}F_i$$
Is there any such meaningful Hilbert space construction for stochastic processes $X(t)$?
probability-theory statistics stochastic-processes hilbert-spaces
edited Jan 31 at 11:37
YuiTo Cheng
2,0392636
asked Jan 23 at 5:29
aghostinthefiguresaghostinthefigures
1,2641217
$endgroup$
It is well-known that one can construct a Hilbert space of zero mean, finite variance random variables such that every random variable of this type $X$ can be represented as a linear combination of a sequence of random variables $F_i$ with respect to probability:
$$X = sum_{i=1}^infty frac{text{cov}(X,F_i)}{text{var}(F_i)}F_i$$
Is there any such meaningful Hilbert space construction for stochastic processes $X(t)$?
probability-theory statistics stochastic-processes hilbert-spaces
probability-theory statistics stochastic-processes hilbert-spaces
edited Jan 31 at 11:37
YuiTo Cheng
2,0392636
asked Jan 23 at 5:29
aghostinthefiguresaghostinthefigures
1,2641217
edited Jan 31 at 11:37
YuiTo Cheng
2,0392636
asked Jan 23 at 5:29
aghostinthefiguresaghostinthefigures
1,2641217
edited Jan 31 at 11:37
YuiTo Cheng
2,0392636
edited Jan 31 at 11:37
YuiTo Cheng
2,0392636
edited Jan 31 at 11:37
YuiTo Cheng
2,0392636
2,0392636
asked Jan 23 at 5:29
aghostinthefiguresaghostinthefigures
1,2641217
asked Jan 23 at 5:29
aghostinthefiguresaghostinthefigures
1,2641217
asked Jan 23 at 5:29
aghostinthefiguresaghostinthefigures
1,2641217
1,2641217
$begingroup$
Is this what you have in mind? en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem
$endgroup$
– Mike Hawk
Jan 28 at 16:25
$begingroup$
@MikeHawk That’s essentially it—although I expected deterministic coefficients on a sequence of stochastic processes instead of stochastic coefficients on a sequence of deterministic processes, I’ll take what I can get; feel free to put it down as an answer and I’ll give you the bounty!
$endgroup$
– aghostinthefigures
Jan 28 at 16:40
add a comment |
$begingroup$
Is this what you have in mind? en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem
$endgroup$
– Mike Hawk
Jan 28 at 16:25
$begingroup$
@MikeHawk That’s essentially it—although I expected deterministic coefficients on a sequence of stochastic processes instead of stochastic coefficients on a sequence of deterministic processes, I’ll take what I can get; feel free to put it down as an answer and I’ll give you the bounty!
$endgroup$
– aghostinthefigures
Jan 28 at 16:40
$begingroup$
Is this what you have in mind? en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem
$endgroup$
– Mike Hawk
Jan 28 at 16:25
$begingroup$
Is this what you have in mind? en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem
$endgroup$
– Mike Hawk
Jan 28 at 16:25
$begingroup$
@MikeHawk That’s essentially it—although I expected deterministic coefficients on a sequence of stochastic processes instead of stochastic coefficients on a sequence of deterministic processes, I’ll take what I can get; feel free to put it down as an answer and I’ll give you the bounty!
$endgroup$
– aghostinthefigures
Jan 28 at 16:40
$begingroup$
@MikeHawk That’s essentially it—although I expected deterministic coefficients on a sequence of stochastic processes instead of stochastic coefficients on a sequence of deterministic processes, I’ll take what I can get; feel free to put it down as an answer and I’ll give you the bounty!
$endgroup$
– aghostinthefigures
Jan 28 at 16:40
add a comment |
1 Answer
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Yes, any square-integrable process on a finite interval $[a,b]$ can be written $X_t=sum_i Z_i e_i(t)$ where $Z_i$ are uncorrelated, $e_i$ are an orthonormal basis of $L^2$, and the convergence is in $L^2$. This is called the Karhunen-Loeve decomposition, and much more information can be found on the Wiki page: https://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem
answered Jan 28 at 17:39
Mike HawkMike Hawk
1,595110
$endgroup$
1
$begingroup$
Jumping on this to mention that the stochastic process $X_t$ can also be viewed as a linear operator between $L^2$ and the Hilbert space of mean-$0$ random variables mentioned in the OP. In this framework, the decomposition of $X_t$ into that sum can be thought of as the infinite-dimensional analogue of the singular-value decomposition of the operator
$endgroup$
– whpowell96
Jan 31 at 3:45
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
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active
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votes
$begingroup$
Yes, any square-integrable process on a finite interval $[a,b]$ can be written $X_t=sum_i Z_i e_i(t)$ where $Z_i$ are uncorrelated, $e_i$ are an orthonormal basis of $L^2$, and the convergence is in $L^2$. This is called the Karhunen-Loeve decomposition, and much more information can be found on the Wiki page: https://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem
answered Jan 28 at 17:39
Mike HawkMike Hawk
1,595110
$endgroup$
1
$begingroup$
Jumping on this to mention that the stochastic process $X_t$ can also be viewed as a linear operator between $L^2$ and the Hilbert space of mean-$0$ random variables mentioned in the OP. In this framework, the decomposition of $X_t$ into that sum can be thought of as the infinite-dimensional analogue of the singular-value decomposition of the operator
$endgroup$
– whpowell96
Jan 31 at 3:45
add a comment |
$begingroup$
Yes, any square-integrable process on a finite interval $[a,b]$ can be written $X_t=sum_i Z_i e_i(t)$ where $Z_i$ are uncorrelated, $e_i$ are an orthonormal basis of $L^2$, and the convergence is in $L^2$. This is called the Karhunen-Loeve decomposition, and much more information can be found on the Wiki page: https://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem
answered Jan 28 at 17:39
Mike HawkMike Hawk
1,595110
$endgroup$
1
$begingroup$
Jumping on this to mention that the stochastic process $X_t$ can also be viewed as a linear operator between $L^2$ and the Hilbert space of mean-$0$ random variables mentioned in the OP. In this framework, the decomposition of $X_t$ into that sum can be thought of as the infinite-dimensional analogue of the singular-value decomposition of the operator
$endgroup$
– whpowell96
Jan 31 at 3:45
add a comment |
$begingroup$
Yes, any square-integrable process on a finite interval $[a,b]$ can be written $X_t=sum_i Z_i e_i(t)$ where $Z_i$ are uncorrelated, $e_i$ are an orthonormal basis of $L^2$, and the convergence is in $L^2$. This is called the Karhunen-Loeve decomposition, and much more information can be found on the Wiki page: https://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem
answered Jan 28 at 17:39
Mike HawkMike Hawk
1,595110
$endgroup$
Yes, any square-integrable process on a finite interval $[a,b]$ can be written $X_t=sum_i Z_i e_i(t)$ where $Z_i$ are uncorrelated, $e_i$ are an orthonormal basis of $L^2$, and the convergence is in $L^2$. This is called the Karhunen-Loeve decomposition, and much more information can be found on the Wiki page: https://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem
answered Jan 28 at 17:39
Mike HawkMike Hawk
1,595110
answered Jan 28 at 17:39
Mike HawkMike Hawk
1,595110
answered Jan 28 at 17:39
Mike HawkMike Hawk
1,595110
answered Jan 28 at 17:39
Mike HawkMike Hawk
1,595110
1,595110
1
$begingroup$
Jumping on this to mention that the stochastic process $X_t$ can also be viewed as a linear operator between $L^2$ and the Hilbert space of mean-$0$ random variables mentioned in the OP. In this framework, the decomposition of $X_t$ into that sum can be thought of as the infinite-dimensional analogue of the singular-value decomposition of the operator
$endgroup$
– whpowell96
Jan 31 at 3:45
add a comment |
1
$begingroup$
Jumping on this to mention that the stochastic process $X_t$ can also be viewed as a linear operator between $L^2$ and the Hilbert space of mean-$0$ random variables mentioned in the OP. In this framework, the decomposition of $X_t$ into that sum can be thought of as the infinite-dimensional analogue of the singular-value decomposition of the operator
$endgroup$
– whpowell96
Jan 31 at 3:45
1
1
$begingroup$
Jumping on this to mention that the stochastic process $X_t$ can also be viewed as a linear operator between $L^2$ and the Hilbert space of mean-$0$ random variables mentioned in the OP. In this framework, the decomposition of $X_t$ into that sum can be thought of as the infinite-dimensional analogue of the singular-value decomposition of the operator
$endgroup$
– whpowell96
Jan 31 at 3:45
$begingroup$
Jumping on this to mention that the stochastic process $X_t$ can also be viewed as a linear operator between $L^2$ and the Hilbert space of mean-$0$ random variables mentioned in the OP. In this framework, the decomposition of $X_t$ into that sum can be thought of as the infinite-dimensional analogue of the singular-value decomposition of the operator
$endgroup$
– whpowell96
Jan 31 at 3:45
add a comment |
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$begingroup$
Is this what you have in mind? en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem
$endgroup$
– Mike Hawk
Jan 28 at 16:25
$begingroup$
@MikeHawk That’s essentially it—although I expected deterministic coefficients on a sequence of stochastic processes instead of stochastic coefficients on a sequence of deterministic processes, I’ll take what I can get; feel free to put it down as an answer and I’ll give you the bounty!
$endgroup$
– aghostinthefigures
Jan 28 at 16:40