Distances and Diameters in Concentration Inequalities: Fromgeometry to Optimal Assignment of Sampling Resources

This note reviews, compares and contrasts three notions of “distance” or “size” that arise often in concentration-ofmeasure inequalities. We review Talagrand’s convex distance and McDiarmid’s diameter, and consider in particular the normal distance on a topological vector space X , which corresponds to the method of Chernoff bounds, and is in some sense “natural” with respect to the duality structure on X . We show that, notably, with respect to this distance, concentration inequalities on the tails of linear, convex, quasiconvex and measurable functions on X are mutually equivalent. We calculate the normal distances that correspond to families of Gaussian and of bounded random variables in R , and to functions ofN empirical means. As an application, we consider the problem of estimating the confidence that one can have in a quantity of interest that depends upon many empirical—as opposed to exact—means and show how the normal distance leads to a formula for the optimal assignment of sampling resources.