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For the conditional covariance matrix, is the notation $\operatorname{cov}(\textbf{X}\mid Y)$ elegimate? where $ \textbf{X} $ is a random vector, and $Y$ is a random variable. I find the definition of conditional covariance matrix for a random vector in the lecture notes of PSU, but is there any assumption before the definition? (e.g. is there any requirement for $ \textbf{X}$ and $ Y $?) And how could I find the accurate definition in a publication? Thank you!

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  • $\begingroup$ What's the matter with the definition supplied in those lecture notes? It looks perfectly standard and rigorous. In what way do you deem it not "accurate"? $\endgroup$
    – whuber
    Sep 1, 2017 at 15:59
  • $\begingroup$ It may help to note that X|Y is itself a random variable, so you can find its covariance like any other rv. $\endgroup$
    – tkmckenzie
    Sep 1, 2017 at 16:12

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There is no problems with the definitions, in fact, $X \mid Y$ is itself a random variable so any definitions for random variables can be applied to it. See here: Interpretation of Total Law of Covariance for use and Wikipedia for other use.

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