%0 Journal Article
%A Sullivan, Tim
%A Owhadi, Houman
%D 2012
%I Begell House
%K concentration of measure, large deviations, normal distance, optimal sampling, Talagrand distance, uncertainty quantification
%N 1
%P 21-38
%R 10.1615/Int.J.UncertaintyQuantification.v2.i1.30
%T DISTANCES AND DIAMETERS IN CONCENTRATION INEQUALITIES: FROM GEOMETRY TO OPTIMAL ASSIGNMENT OF SAMPLING RESOURCES
%U http://dl.begellhouse.com/journals/52034eb04b657aea,69f226067bce0f5b,721bdd650e6dde89.html
%V 2
%X 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
ℝ^{N}, 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.
%8 2012-03-07