I am a computer science guy, not a mathematician so kindly excuse me if there is any ridiculous error in my problem description.

I have two clusters $C_1$ and $C_2$ in a feature space spanned by $k$ features scaled between $0$ and $1$. That means each feature ranges between $0$ and $1$ after min-max normalization of the feature values. The clusters are generated while sampling from two separate multivariate Gaussian distributions $N_1(f_1, f_2, ...f_k )$ and $N_2(f_1, f_2, ...f_n )$ where $f_i \in [0,1]$, over the scaled features. I do not have a condition on means or covariance matrices. However it is given that any pair of vectors (or instances) within a given cluster should not have the Euclidean distance more than a specified scaler $D$. Now two arbitrary vectors $A$ and $B$ are picked up from clusters $C_1$ and $C_2$ respectively. Now $R_A$ and $R_B$ are the feature orderings based on their corresponding values in the vectors (for example, if $A$=(0.8,0.2,0.7), $R_A$ is an ordered list $(f_2<f_3<f_1)$). What is the probability that Kendall's tau distance between $R_A$ and $R_B$ is $0$. I need an expression for $Pr(K(R_A,R_B)=0 | D=d)$.

I hope I could explain the problem unambiguously. However, kindly let me know if there is any confusion.




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