If $X$ and $Y$ are independent random variables, then their Spearman's Correlation Coefficient is...? I have often seen the following statement:

If $X$ and $Y$ are independent random variables, then their 'correlation' is $0$

What is meant by 'correlation'? Is it specific to measures of linear correlation? Or does this claim apply to non-linear measures of correlation, as well?  (e.g. if $X$ and $Y$ are independent variables, then their Spearman's Correlation Coefficient is...?)

Let $X, Y$ be independent random variables. Let $r()$ be the rank transformation. Then the Spearman correlation is the same as the Pearson correlation of the ranks, that is, of $r(X), r(Y)$.
Since Functions of Independent Random Variables are independent, $r(X)$ and $r(Y)$ are independent, so their Pearson correlation is zero, which is to say that the Spearman correlation of $X$ and $Y$ is zero.
