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Suppose that there are some data collected in the form of time series from 3 subjects:

$X_1 = (x_{11}, x_{12}, ..., x_{1n})$

$X_2 = (x_{21}, x_{22}, ..., x_{2n})$

$X_3 = (x_{31}, x_{32}, ..., x_{3n})$

To simplify the situation, assume that the components in each time series are independent from each other and follow $N(0, \sigma_1^2)$, $N(0, \sigma_2^2)$, and $N(0, \sigma_3^2)$ respectively. The Pearson correlation coefficients among the three subjects are,

$r_{12} = E(X_1 X_2)/(\sigma_1 \sigma_2)$

$r_{13} = E(X_1 X_3)/(\sigma_1 \sigma_3)$

$r_{23} = E(X_2 X_3)/(\sigma_2 \sigma_3)$

My question is, are the 3 correlation coefficients (as random variables) independent from each other? If not, how to demonstrate the situation?

bluepole
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    http://stats.stackexchange.com/questions/72790 gives restrictions on the three correlation coefficients. Since the coefficients separately are subject to no such restrictions (apart from the mathematically imposed one that they lie between $-1$ and $1$), this immediately implies lack of independence. – whuber Feb 20 '15 at 23:19
  • @whuber If the restriction within -1 and 1 is an issue, how about rephrasing the question: if we Fisher transform the 3 correlation coefficients to z-values, $z_{11}, z_{12}, z_{13}$, are the 3 z-scores independent from each other? – bluepole Feb 21 '15 at 00:00
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    No, for the same reason. – whuber Feb 21 '15 at 04:20
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    @whuber I read that thread carefully, and am convinced by the conclusion. My intuition did tell me they should not be independent from each other. The twist is the following. I computed the correlations among the randomly generated time series from $n$ subjects, and ran some simulations to assess the controllability of false positives at a nominal significance level (e.g., 0.05) for a one-sample $t$-test on the $n(n+1)/2$ z-values of Fisher-transformed correlations. I expected to see higher false positives due to the dependency, but was surprised that didn't occur! What am I missing here? – bluepole Feb 21 '15 at 16:09
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    That is an interesting question. If you don't mind, please consider amplifying it by means of an edit to this post: that should get the attention of many people who might be able to add to your insight. – whuber Feb 21 '15 at 18:50

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