102 research outputs found
Uniform convergence of Vapnik--Chervonenkis classes under ergodic sampling
We show that if is a complete separable metric space and
is a countable family of Borel subsets of with
finite VC dimension, then, for every stationary ergodic process with values in
, the relative frequencies of sets 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 . 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.Comment: Published in at http://dx.doi.org/10.1214/09-AOP511 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Denoising Deterministic Time Series
This paper is concerned with the problem of recovering a finite,
deterministic time series from observations that are corrupted by additive,
independent noise. A distinctive feature of this problem is that the available
data exhibit long-range dependence and, as a consequence, existing statistical
theory and methods are not readily applicable. This paper gives an analysis of
the denoising problem that extends recent work of Lalley, but begins from first
principles. Both positive and negative results are established. The positive
results show that denoising is possible under somewhat restrictive conditions
on the additive noise. The negative results show that, under more general
conditions on the noise, no procedure can recover the underlying deterministic
series
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