On the Non-degeneracy of Kendall’s Correlation Coefficient

Let (X,Y ) be a random point in R with a (joint) cumulative distribution function (c.d.f.) F and continuous marginal c.d.f.’s FX and FY , so that F (x, y) = P(X 6 x, Y 6 y), FX(x) = P(X 6 x), and FY (y) = P(Y 6 y) for all real x and y. Then F is continuous as well, since |F (x2, y2)−F (x1, y1)| 6 |FX(x2)−FX(x1)|+|FY (y2)−FY (y1)| for all x1, y1, x2, y2 in R. Vice versa, if F (x, y) is continuous in x for each real y, then 0 = F (x, y)−F (x−, y) = P(X = x, Y 6 y) −→ y→∞ P(X = x) = FX(x) − FX(x−), so that FX is continuous. Similarly, if F (x, y) is continuous in y for each real x, then FY is continuous. So, the marginal c.d.f.’s FX and FY are continuous iff the joint c.d.f. F is so; in such a case, one may simply say that the distribution of (X,Y ) is continuous. Let μ = μX,Y denote the measure that is the probability distribution of (X,Y ). Let S = SX,Y stand for the set in R 2 that is the support of μ (