medium · FRM Part 1 Quantitative Analysis
X and Y are each standard normal with linear correlation ρ = 0. An analyst concludes they are independent and that the tail-dependence in a stress scenario is therefore zero.
Under which single additional assumption is the analyst's independence conclusion guaranteed correct, and why does ρ=0 alone fail?
- Correct only if (X,Y) is jointly (bivariate) normal; zero correlation alone permits dependence such as Y=X^2-type structure where ρ=0 yet X and Y are clearly dependent.
- Correct unconditionally: for standard normal marginal distributions, a zero linear correlation always implies full statistical independence between the two random variables.
- Correct only if X and Y have finite fourth moments; zero correlation combined with finite kurtosis is a mathematically sufficient condition for full statistical independence in this case.
- Correct only if Spearman's rank correlation is also exactly zero; matching Pearson and Spearman correlations of zero together are jointly sufficient to imply independence for any margins here.
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