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Post Closed as "Duplicate" by whuber♦ self-studyUsers with the self-study badge or a synonym can single-handedly close self-study questions as duplicates and reopen them as needed.
Suppose $\mathbf{X, Y}$ are independent random vectors. Are their components independent?
Let $\mathbf{X} = (X_1, \dots, X_p)^\top$ and $\mathbf{Y} = (Y_1, \dots, Y_p)^\top$ be independent. Does it then follow that $X_i$ is independent with $Y_j$ i.e. cov$(X_i, Y_j) = 0$?
