If we have a diagonal covariance matrix does that guarantee independency?
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No, not in general. A covariance of zero between two random variables does not necessarily imply that they are independent. However, the statement is true if the variables are normally distributed.
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In fact, zero covariance does not imply independence even for normally distributed variables. See [here](https://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent#A_symmetric_example). Rather, if two variables are from a multivariable normal distribution and are zero covariance, it implies independence. – Tom Bennett Feb 04 '21 at 22:42