Four simple axioms of dependence measures (Q1731093)

The starting point of this paper is the list of seven axioms proposed by \textit{A. Rényi} [Acta Math. Acad. Sci. Hung. 10, 441--451 (1959; Zbl 0091.14403)]; it is known that only maximal correlation satisfies all Rényi's axioms, altough they do not characterise it. The authors propose for a pair of random variables $(X,Y)$ a ``minimalist'' system of axioms as follows: (i) $\Delta(X,Y)=0$ if, and only if, $X$ and $Y$ are independent, (ii) $\Delta(X,Y)=\Delta(LX,MY)$ for all similarity transformations $L$ and $M$, (iii) $\Delta(X,Y)=1$ if, and only if, $Y=LX$ for a similarity transformation, (iv) $\Delta(X,Y)$ is continuous with respect to weak convergence. The distance correlation introduced by \textit{G. J. Székely} et al. [Ann. Stat. 35, No. 6, 2769--2794 (2007; Zbl 1129.62059)] satisfies all the axioms (i)--(iv). Axiom (iv) is especially important since many measures of correlation do not satisfy it.