I have the following question:
If two random vectors $X$ and $Y$ have the same distribution and $X$ has independent components, does $Y$ have also independent components?
If it is not true, why is it true when $X$ is a $d$-Gaussian vector?
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Sign up to join this communityI have the following question:
If two random vectors $X$ and $Y$ have the same distribution and $X$ has independent components, does $Y$ have also independent components?
If it is not true, why is it true when $X$ is a $d$-Gaussian vector?
$\quad \mathbb P(Y_1 \le y_1, Y_2 \le y_2, \ldots , Y_n \le y_n)$ the CDF of $\mathbf Y$
$=\mathbb P(X_1 \le y_1, X_2 \le y_2, \ldots , X_n \le y_n)$ since $\mathbf{X}$ and $\mathbf Y$ have the same distribution
$=\mathbb P(X_1 \le y_1) \mathbb P(X_2 \le y_2) \cdots \mathbb P(X_n \le y_n)$ by independence of the components of $\mathbf X$
$=\mathbb P(Y_1 \le y_1) \mathbb P(Y_2 \le y_2) \cdots \mathbb P(Y_n \le y_n)$ since $\mathbf{X}$ and $\mathbf Y$ have the same distribution
So the components of $\mathbf Y$ are also independent and this does not depend on the shapes of the individual marginal distributions of the components