Pearson Coefficient Degradation in a Wasserstein/Gelbrich Ambiguity Set

This letter presents new results on the Gelbrich distance and the corresponding ambiguity sets, to analyze the correlation between two scalar random variables. A closed expression of the minimum disturbance in the Gelbrich metric necessary to achieve a specified correlation between two random variables is proposed. This expression allows us to analytically compute the minimal absolute correlation in a Gelbrich ball. This analysis facilitates the assessment of the robustness of the Pearson coefficient within an ambiguity set. Two numerical examples illustrate the validity of the proposed results.