RT Journal Article ID 721bdd650e6dde89 A1 Sullivan, Tim A1 Owhadi, Houman T1 DISTANCES AND DIAMETERS IN CONCENTRATION INEQUALITIES: FROM GEOMETRY TO OPTIMAL ASSIGNMENT OF SAMPLING RESOURCES JF International Journal for Uncertainty Quantification JO IJUQ YR 2012 FD 2012-03-07 VO 2 IS 1 SP 21 OP 38 K1 concentration of measure K1 large deviations K1 normal distance K1 optimal sampling K1 Talagrand distance K1 uncertainty quantification AB This note reviews, compares and contrasts three notions of "distance" or "size" that arise often in concentration-of-measure 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 ℝN, and to functions of N 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. PB Begell House LK https://www.dl.begellhouse.com/journals/52034eb04b657aea,69f226067bce0f5b,721bdd650e6dde89.html