Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling
Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling
复制标题
遍历采样下 Vapnik-Chervonenkis 类的均匀收敛
DOI：
10.1214/09-aop511
复制
发表时间：
2010
影响因子：
2.3
通讯作者：
A. Nobel
中科院分区：
文献类型：
--
作者：
Terrence M. Adams;A. Nobel
我们表明，如果x是一个完整的单独的度量空间，而C是具有有限VC维度的X的borel子集的一个可数家族，那么，对于每个固定的ergodic过程，x中的值，x中的值，集合的相对频率c∈Ccomemengge统一的限制可能性。并强烈混合过程。作为集合的基本结果的推论。
We show that if X is a complete separable metric space and C is a countable family of Borel subsets of X with finite VC dimension, then, for every stationary ergodic process with values in X, the relative frequencies of sets C ∈ C converge uniformly to their limiting probabilities. Beyond ergodicity, no assumptions are imposed on the sampling process, and no regularity conditions are imposed on the elements of C. The result extends existing work of Vapnik and Chervonenkis, among others, who have studied uniform convergence for i.i.d. and strongly mixing processes. Our method of proof is new and direct: it does not rely on symmetrization techniques, probability inequalities or mixing conditions. The uniform convergence of relative frequencies for VC-major and VC-graph classes of functions under ergodic sampling is established as a corollary of the basic result for sets.