2004/07/01 by Shahar Mendelson, Gideon Schechtman
Mathematics · #Geometry and complex manifolds #Numerical methods in inverse problems #Point processes and geometric inequalities #math.PR #msc:46B09. #msc:60D05
paper · pdf · doi:10.1214/009117904000000388
published as Annals of Probability 2004, Vol. 32, No. 3A, 1746-1770 · Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000388
openalex publication_date 2004/07/01 · arxiv created 2004/10/05 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We evaluate the shattering dimension of various classes of linear functionals on various symmetric convex sets. The proofs here relay mostly on methods from the local theory of normed spaces and include volume estimates, factorization techniques and tail estimates of norms, viewed as random variables on Euclidean spheres. The estimates of shattering dimensions can be applied to obtain error bounds for certain classes of functions, a fact which was the original motivation of this study. Although this can probably be done in a more traditional manner, we also use the approach presented here to determine whether several classes of linear functionals satisfy the uniform law of large numbers and the uniform central limit theorem.