Some comments on concentration and expansion functions as applied to bivariate dependence (Q5890268)

The present paper surveys and investigates various definitions and approaches to the notions of concentration functions (CFs) and expansion functions (EFs) for two random variables. The main problem is to examine the analytical properties of CFs and EFs, as well as the relationship between concentration-expansion functions and the dependence relation between the two random variables. The authors emphasise the following concluding remarks:NEWLINENEWLINENEWLINE(1) If CFs and EFs are to be used for statistical applications on bivariate dependence, they must be both used since CFs seem to be appropriate for negative dependence, while EFs appear to be suitable for positive dependence. CFs and EFs are nonetheless incapable of detecting complete dependencies, especially positive complete dependence.NEWLINENEWLINENEWLINE(2) Another important conclusion is that the notions of concentration and expansion are intrinsically not identical with the concept of dependence bearing on two random variables. Concentration is related to detection of continuity and singularity of two probability measures, while expansion bears on the question whether joint probabilities (distributions) are formed by the product of their marginal measures (distributions) or not.NEWLINENEWLINENEWLINE(3) Since both martingales and joint measures are defined on a common measurable space, joint measures (distributions) cannot be singular with respect to the product of their marginals (marginal distributions) everywhere in the relevant space (unless for the empty probability space).