Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling

@article{Adams2010UniformCO,
  title={Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling},
  author={Terrence M. Adams and Andrew B. Nobel},
  journal={Annals of Probability},
  year={2010},
  volume={38},
  pages={1345-1367}
}
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… 
Uniform Approximation of Vapnik-Chervonenkis Classes
For any family of measurable sets in a probability space, we show that either (i) the family has infinite Vapnik-Chervonenkis (VC) dimension or (ii) for every epsilon > 0 there is a finite partition
The Gap Dimension and Uniform Laws of Large Numbers for Ergodic Processes
Let F be a family of Borel measurable functions on a complete separable metric space. The gap (or fat-shattering) dimension of F is a combinatorial quantity that measures the extent to which
Sequential complexities and uniform martingale laws of large numbers
We establish necessary and sufficient conditions for a uniform martingale Law of Large Numbers. We extend the technique of symmetrization to the case of dependent random variables and provide
Entropy and the uniform mean ergodic theorem for a family of sets
We define a notion of entropy for an infinite family $\mathcal{C}$ of measurable sets in a probability space. We show that the mean ergodic theorem holds uniformly for $\mathcal{C}$ under every
A counterexample concerning the extension of uniform strong laws to ergodic processes
We present a construction showing that a class of sets C that is Glivenko-Cantelli for an i.i.d. process need not be Glivenko-Cantelli for every stationary ergodic process with the same one
Empirical Processes, Typical Sequences, and Coordinated Actions in Standard Borel Spaces
  • M. Raginsky
  • Computer Science, Mathematics
    IEEE Transactions on Information Theory
  • 2013
This paper proposes a new notion of typical sequences on a wide class of abstract alphabets (so-called standard Borel spaces), which is based on approximations of memoryless sources by empirical
Generalization Bounds for Minimum Volume Set Estimation based on Markovian Data
The main goal of this paper is to establish generalization bounds for minimum volume set estimation for regenerative Markov chains. We obtain new maximal concentration inequality in order to show
Uniform Approximation and Bracketing Properties of VC classes
We show that the sets in a family with finite VC dimension can be uniformly approximated within a given error by a finite partition. Immediate corollaries include the fact that VC classes have finite
Bootstrap and uniform bounds for Harris Markov chains
This thesis concentrates on some extensions of empirical processes theory when the data are Markovian. More specifically, we focus on some developments of bootstrap, robustness and statistical
TOPICS IN COMBINATORICS (MATH 285N, UCLA, WINTER 2016)
Let X be a set (finite or infinite), and let F be a family of subsets of X. A pair (X,F) is called a set system. Given A ⊆ X, we say that the family F shatters A if for every A′ ⊆ A, there is a set S
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 45 REFERENCES
Uniform Ergodic Theorems for Dynamical Systems Under VC Entropy Conditions
The classic limit theorems of Vapnik and Chervonenkis [28, 29] show that if a function class F satisfies a random entropy condition, then the strong law of large numbers holds uniformly over F. In
Uniform Ergodic Theorems for Dynamical Systems Under VC Entropy Conditions
The classic limit theorems of Vapnik and Chervonenkis [27,28] show that if a function class F satisfies a random entropy condition, then the strong law of large numbers holds uniformly over F . In
The Glivenko-Cantelli Problem
We give a new type of characterization of the Glivenko-Cantelli classes. In the case of a class $\mathscr{L}$ of sets, the characterization is closely related to the configuration that the sets of
Necessary and Sufficient Conditions for the Uniform Law of Large Numbers in the Stationary Case
Necessary and sufficient conditions for the uniform law of large numbers for stationary ergodic sequences of random variables are given. Three different types of conditions are investigated and
RATES OF CONVERGENCE FOR EMPIRICAL PROCESSES OF STATIONARY MIXING SEQUENCES
Classical empirical process theory for Vapnik-Cervonenkis classes deals mainly with sequences of independent variables. This paper extends the theory to stationary sequences of dependent variables.
The Uniform Mean-Square Ergodic Theorem for Wide Sense Stationary Processes
It is shown that the uniform mean-square ergodic theorem holds for the family of wide sense stationary sequences, as soon as the random process with orthogonal increments, which corresponds to the
Measure Theory
These are some brief notes on measure theory, concentrating on Lebesgue measure on Rn. Some missing topics I would have liked to have included had time permitted are: the change of variable formula
What is ergodic theory
Ergodic theory involves the study of transformations on measure spaces. Interchanging the words “measurable function” and “probability density function” translates many results from real analysis to
Probability measures on metric spaces
...
1
2
3
4
5
...