- - - on Characterizing Dependence in Joint Distributions

Ways of characterizing the dependence of one random variable on another (or several others) are investigated. In particular, an index of dependence of X on Y is introduced which (i) always eXists, (ii) lies between zero and unity inclusive, (iii) is zero if and only if X and Yare independent, (iv) is unity if X is a function of Y (and only if whenever X has finite variance), (v) may assume every value between zero and unity inclusive by varying the joint distribution but holding the marginal distributions fixed (assuming Y continuously distributed), (vi) is invariant under linear transformation of X and one-to-one transformation of Y, and (vii) equals kim whenever X and Yare sums of (non-degenerate) independent and identically distributed random variables Zl,Z2' • • • 'X being the sum of the first m ZI S and Y the sum of therirst k ZI S (m > k). When the correlation ratio eXists, its square cannot exceed the dependence index, and when (X,Y) is either bivariate normal or trinomial in distribution then the index equals the square of the correlation coefficient. The index is derived by first introducing and investigating a dependence characteristic, defined as the correlation ratio of exp(itX) on Y as a function of t. A correlation characteristic and index are also introduced. A brief survey of correlation and regression theory for complex-valued random variables is included. (No statistical aspects of dependence are considered).