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Using the change-of-variables formula for pushforward measures for $Bbb P_X = Bbb P_-X Rightarrow varphi_X = varphi_-X$
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Let $X$ be $Bbb R^d$-valued random variable with $varphi_X$ is characteristic function and $Bbb P_X$ its distribution. I want to use the change-of-variables formula for pushforward measures ($ast$) to show that $Bbb P_X = Bbb P_-X Rightarrow varphi_X = varphi_-X$.
$varphi_-X(t) = varphi_X(-t) = int_Bbb R^d e^i langle -t,s rangle dBbb P_X[ds] = int_Bbb R^d phi circ A dBbb P_X[ds]$ where $phi: Bbb R^d to Bbb R, s mapsto e^i langle t,-s rangle$ and $A = -I_d$.
Then, $int_Bbb R^d phi circ A dBbb P_X[ds] = int_Bbb R^d phi circ A dBbb P_X[ds circ A^-1]$ by $(ast)$.
Using that $A^-1 = A$ and $Bbb P_X(-A) = Bbb P_X(A)$ for all $A in mathcal B^d$ gives us the result.
probability-theory proof-verification
asked Jul 15 at 0:09
Pazu
359213
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up vote
1
down vote
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Let $X$ be $Bbb R^d$-valued random variable with $varphi_X$ is characteristic function and $Bbb P_X$ its distribution. I want to use the change-of-variables formula for pushforward measures ($ast$) to show that $Bbb P_X = Bbb P_-X Rightarrow varphi_X = varphi_-X$.
$varphi_-X(t) = varphi_X(-t) = int_Bbb R^d e^i langle -t,s rangle dBbb P_X[ds] = int_Bbb R^d phi circ A dBbb P_X[ds]$ where $phi: Bbb R^d to Bbb R, s mapsto e^i langle t,-s rangle$ and $A = -I_d$.
Then, $int_Bbb R^d phi circ A dBbb P_X[ds] = int_Bbb R^d phi circ A dBbb P_X[ds circ A^-1]$ by $(ast)$.
Using that $A^-1 = A$ and $Bbb P_X(-A) = Bbb P_X(A)$ for all $A in mathcal B^d$ gives us the result.
probability-theory proof-verification
asked Jul 15 at 0:09
Pazu
359213
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $X$ be $Bbb R^d$-valued random variable with $varphi_X$ is characteristic function and $Bbb P_X$ its distribution. I want to use the change-of-variables formula for pushforward measures ($ast$) to show that $Bbb P_X = Bbb P_-X Rightarrow varphi_X = varphi_-X$.
$varphi_-X(t) = varphi_X(-t) = int_Bbb R^d e^i langle -t,s rangle dBbb P_X[ds] = int_Bbb R^d phi circ A dBbb P_X[ds]$ where $phi: Bbb R^d to Bbb R, s mapsto e^i langle t,-s rangle$ and $A = -I_d$.
Then, $int_Bbb R^d phi circ A dBbb P_X[ds] = int_Bbb R^d phi circ A dBbb P_X[ds circ A^-1]$ by $(ast)$.
Using that $A^-1 = A$ and $Bbb P_X(-A) = Bbb P_X(A)$ for all $A in mathcal B^d$ gives us the result.
probability-theory proof-verification
asked Jul 15 at 0:09
Pazu
359213
Let $X$ be $Bbb R^d$-valued random variable with $varphi_X$ is characteristic function and $Bbb P_X$ its distribution. I want to use the change-of-variables formula for pushforward measures ($ast$) to show that $Bbb P_X = Bbb P_-X Rightarrow varphi_X = varphi_-X$.
$varphi_-X(t) = varphi_X(-t) = int_Bbb R^d e^i langle -t,s rangle dBbb P_X[ds] = int_Bbb R^d phi circ A dBbb P_X[ds]$ where $phi: Bbb R^d to Bbb R, s mapsto e^i langle t,-s rangle$ and $A = -I_d$.
Then, $int_Bbb R^d phi circ A dBbb P_X[ds] = int_Bbb R^d phi circ A dBbb P_X[ds circ A^-1]$ by $(ast)$.
Using that $A^-1 = A$ and $Bbb P_X(-A) = Bbb P_X(A)$ for all $A in mathcal B^d$ gives us the result.
probability-theory proof-verification
asked Jul 15 at 0:09
Pazu
359213
asked Jul 15 at 0:09
Pazu
359213
asked Jul 15 at 0:09
Pazu
359213
asked Jul 15 at 0:09
Pazu
359213
359213
add a comment |Â
add a comment |Â
1 Answer
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If $mathbb P_X=mathbb P_-X$ then $Ef(X)=Ef(-X)$ for any bounded measurable function $f$. Take $f(x)=e^itx$ to get $phi_X=phi_-X$. Your argument is correct but unnecessary.
answered Jul 15 at 0:20
Kavi Rama Murthy
21k2830
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
If $mathbb P_X=mathbb P_-X$ then $Ef(X)=Ef(-X)$ for any bounded measurable function $f$. Take $f(x)=e^itx$ to get $phi_X=phi_-X$. Your argument is correct but unnecessary.
answered Jul 15 at 0:20
Kavi Rama Murthy
21k2830
add a comment |Â
up vote
1
down vote
accepted
If $mathbb P_X=mathbb P_-X$ then $Ef(X)=Ef(-X)$ for any bounded measurable function $f$. Take $f(x)=e^itx$ to get $phi_X=phi_-X$. Your argument is correct but unnecessary.
answered Jul 15 at 0:20
Kavi Rama Murthy
21k2830
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
If $mathbb P_X=mathbb P_-X$ then $Ef(X)=Ef(-X)$ for any bounded measurable function $f$. Take $f(x)=e^itx$ to get $phi_X=phi_-X$. Your argument is correct but unnecessary.
answered Jul 15 at 0:20
Kavi Rama Murthy
21k2830
If $mathbb P_X=mathbb P_-X$ then $Ef(X)=Ef(-X)$ for any bounded measurable function $f$. Take $f(x)=e^itx$ to get $phi_X=phi_-X$. Your argument is correct but unnecessary.
answered Jul 15 at 0:20
Kavi Rama Murthy
21k2830
answered Jul 15 at 0:20
Kavi Rama Murthy
21k2830
answered Jul 15 at 0:20
Kavi Rama Murthy
21k2830
answered Jul 15 at 0:20
Kavi Rama Murthy
21k2830
21k2830
add a comment |Â
add a comment |Â
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