The shattering dimension of sets of linear functionals

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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.
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

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Probability, 60D05, 46B09. (Primary)

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http://salesiana.dossiersoluciones.com/handle/123456789/72742

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0410096
 http://arxiv.org/abs/math/0410096
 Annals of Probability 2004, Vol. 32, No. 3A, 1746-1770
 doi:10.1214/009117904000000388

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arXiv