Asymptotic representations for Spearman's footrule correlation coefficient

In order to address the theoretical challenges arising from the dependence structure of ranks in Spearman's footrule correlation coefficient, we propose two asymptotic representations to approximate the distribution of this coefficient under the hypothesis of independence. The first representation simplifies the dependence structure by replacing empirical distribution functions with their population counterparts. The second representation leverages the H\'{a}jek projection technique to decompose the initial form into a sum of independent components, thereby rigorously justifying asymptotic normality. Simulation studies demonstrate the appropriateness of two proposed asymptotic representations, as well as their excellent approximation to the limiting normal distribution.