Mixing
Show that X is a sub-gaussian random vector with dependent sub-gaussian coordinates
$begingroup$
Let $X in R^n$ be a zero mean, random vector with sub-gaussian coordinates $X_i$.
prove that X is a sub-gaussian random vector no matter if coordinates are independent or dependent.
It is easy to prove the result in the case of independent coordinates.
When it comes to the case of dependent coordinates, I think of the definition of multivariate normal distribution but don't know if it works for sub-gaussian family.
Assume a random vector Z $in R^n$ has independent zero mean, unit variance, sub-gaussian coordinates and denote $Sigma_X$as the covariance matrix of X, we can find a Z such that $Sigma_X^{1/2} Z$ has the same distribution as X.
Because for $forall a in R^n$ $a^{T}Sigma_X^{1/2} in R^n$, given the case of independent coordinates, we can say for $forall a in R^n, a^{T}Sigma_X^{1/2}Z = a^{T}X$ is sub-gaussion distributed. So X is a sub-gaussian random vector.
I am not sure if the proof is right for the whole sub-gaussian family, because I am not sure we can find a Z that $Sigma_X^{1/2} Z$ is distributed same as X.
Any suggestions and ideas?
probability statistics normal-distribution
asked Jan 13 at 18:32
DylonDylon
63
$endgroup$
add a comment |
$begingroup$
Let $X in R^n$ be a zero mean, random vector with sub-gaussian coordinates $X_i$.
prove that X is a sub-gaussian random vector no matter if coordinates are independent or dependent.
It is easy to prove the result in the case of independent coordinates.
When it comes to the case of dependent coordinates, I think of the definition of multivariate normal distribution but don't know if it works for sub-gaussian family.
Assume a random vector Z $in R^n$ has independent zero mean, unit variance, sub-gaussian coordinates and denote $Sigma_X$as the covariance matrix of X, we can find a Z such that $Sigma_X^{1/2} Z$ has the same distribution as X.
Because for $forall a in R^n$ $a^{T}Sigma_X^{1/2} in R^n$, given the case of independent coordinates, we can say for $forall a in R^n, a^{T}Sigma_X^{1/2}Z = a^{T}X$ is sub-gaussion distributed. So X is a sub-gaussian random vector.
I am not sure if the proof is right for the whole sub-gaussian family, because I am not sure we can find a Z that $Sigma_X^{1/2} Z$ is distributed same as X.
Any suggestions and ideas?
probability statistics normal-distribution
asked Jan 13 at 18:32
DylonDylon
63
$endgroup$
add a comment |
$begingroup$
Let $X in R^n$ be a zero mean, random vector with sub-gaussian coordinates $X_i$.
prove that X is a sub-gaussian random vector no matter if coordinates are independent or dependent.
It is easy to prove the result in the case of independent coordinates.
When it comes to the case of dependent coordinates, I think of the definition of multivariate normal distribution but don't know if it works for sub-gaussian family.
Assume a random vector Z $in R^n$ has independent zero mean, unit variance, sub-gaussian coordinates and denote $Sigma_X$as the covariance matrix of X, we can find a Z such that $Sigma_X^{1/2} Z$ has the same distribution as X.
Because for $forall a in R^n$ $a^{T}Sigma_X^{1/2} in R^n$, given the case of independent coordinates, we can say for $forall a in R^n, a^{T}Sigma_X^{1/2}Z = a^{T}X$ is sub-gaussion distributed. So X is a sub-gaussian random vector.
I am not sure if the proof is right for the whole sub-gaussian family, because I am not sure we can find a Z that $Sigma_X^{1/2} Z$ is distributed same as X.
Any suggestions and ideas?
probability statistics normal-distribution
asked Jan 13 at 18:32
DylonDylon
63
$endgroup$
Let $X in R^n$ be a zero mean, random vector with sub-gaussian coordinates $X_i$.
prove that X is a sub-gaussian random vector no matter if coordinates are independent or dependent.
It is easy to prove the result in the case of independent coordinates.
When it comes to the case of dependent coordinates, I think of the definition of multivariate normal distribution but don't know if it works for sub-gaussian family.
Assume a random vector Z $in R^n$ has independent zero mean, unit variance, sub-gaussian coordinates and denote $Sigma_X$as the covariance matrix of X, we can find a Z such that $Sigma_X^{1/2} Z$ has the same distribution as X.
Because for $forall a in R^n$ $a^{T}Sigma_X^{1/2} in R^n$, given the case of independent coordinates, we can say for $forall a in R^n, a^{T}Sigma_X^{1/2}Z = a^{T}X$ is sub-gaussion distributed. So X is a sub-gaussian random vector.
I am not sure if the proof is right for the whole sub-gaussian family, because I am not sure we can find a Z that $Sigma_X^{1/2} Z$ is distributed same as X.
Any suggestions and ideas?
probability statistics normal-distribution
probability statistics normal-distribution
asked Jan 13 at 18:32
DylonDylon
63
asked Jan 13 at 18:32
DylonDylon
63
edited Jan 15 at 1:36
Dylon
asked Jan 13 at 18:32
DylonDylon
63
asked Jan 13 at 18:32
DylonDylon
63
asked Jan 13 at 18:32
DylonDylon
63
63
add a comment |
add a comment |
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