If we have i.i.d. random variables, $X$ and $Y$, then $\text{Cov}(X,Y)=0$.
But let's say we have i.i.d. random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$, where $\boldsymbol{X}=(X_{1},...,X_{p})$ and $\boldsymbol{Y}=(Y_{1},...,Y_{q})$. Do we have any properties analogous to the univariate case: $\text{Cov}(X,Y)=0$, that always hold true, as a result of these random vectors being i.i.d.? Something like $\text{Cov}(X_i,X_j)=0$ if $i\neq j$, or $\text{Cov}(X_i,Y_i)=0$, or maybe the independence of $\boldsymbol{X}$ and $\boldsymbol{Y}$ gives a covariance matrix with zero terms everywhere except the diagonal (these are all just guesses, and not necessarily true). Are there any properties like this for i.i.d. random vectors?