Let's say we have i.i.d. normal distributions $Z_1, ..., Z_n$. For some nonzero constants $a_1, \cdots, a_n$ and $b_1, \cdots, b_n$, if $X = a_{1}Z_{1} + \cdots + a_{n}Z_{n}$ and $Y = b_{1}Z_{1} + \cdots + b_{n}Z_{n}$ have covariance zero, are X and Y necessarily independent?

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    $\begingroup$ I'm pretty sure this is effectively a duplicate. This one addresses the central issue (that (X,Y) will be bivariate normal), from which the answer to your question follows (if that's not obvious see here), but I think there's another one which even more directly addresses this.question $\endgroup$ – Glen_b -Reinstate Monica Mar 3 '17 at 8:26
  • $\begingroup$ Yes, $X$ and $Y$ will have a bivariate normal distribution, therefore zero covariance implies independence. $\endgroup$ – gammer Mar 5 '17 at 3:57

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