Small deviations of smooth stationary Gaussian processes

@article{FAurzada2008SmallDO,
  title={Small deviations of smooth stationary Gaussian processes},
  author={F.Aurzada and I.A.Ibragimov and M.A.Lifshits and J. H. van Zanten},
  journal={Theory of Probability and Its Applications},
  year={2008},
  volume={53},
  pages={697-707}
}
We investigate the small deviation probabilities of a class of very smooth stationary Gaussian processes playing an important role in Bayesian statistical inference. Our calculations are based on the appropriate modification of the entropy method due to Kuelbs, Li, and Linde as well as on classical results about the entropy of classes of analytic functions. They also involve Tsirelson's upper bound for small deviations and shed some light on the limits of sharpness for that estimate. 

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References

SHOWING 1-10 OF 18 REFERENCES

Bayesian inference with rescaled Gaussian process priors

TLDR
This work exhibits rescaled Gaussian process priors yielding posteriors that contract around the true parameter at optimal convergence rates and establishes bounds on small deviation probabilities for smooth stationary Gaussian processes.

Small deviations of series of independent positive random variables with weights close to exponential

Let ξ, ξ0, ξ1, ... be independent identically distributed (i.i.d.) positive random variables. The present paper is a continuation of the article [1] in which the asymptotics of probabilities of small

Approximation, metric entropy and small ball estimates for Gaussian measures

TLDR
This work relates the small ball behavior of a Gaussian measure μ on a Banach space E with the metric entropy behavior of K μ, the unit ball of the reproducing kernel Hilbert space of μ in E to enable the application of tools and results from functional analysis to small ball problems.

Small Deviation Probabilities of Sums of Independent Random Variables

Let ξ1,ξ2, … be a sequence of independent N(0, l)-distributed random variables (r.v.’s) and let (o(j))∞j=1 be a summable sequence of positive real numbers. The sum S := ∑∞j=1 o(j)ξ2 jis then well

Metric entropy and the small ball problem for Gaussian measures

Abstract We establish a precise link between the small ball problem for a Gaussian measure μ on a separable Banach space and the metric entropy of the unit ball of the Hubert space H μ generating μ.

Gaussian Random Functions

TLDR
This book discusses Gaussian distributions and random variables, the functional law of the iterated logarithm, and several open problems.

Asymptotic behavior of the eigenvalues of certain integral equations

then To is just convolution by k. However we shall not insist that this be the case. Let V(x) be a bounded non-negative function with bounded support and denote by MVI/2 the operator on L2(Ed) which

Gaussian Processes for Machine Learning

TLDR
The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics, and deals with the supervised learning problem for both regression and classification.