New Bernstein and Hoeffding type inequalities for regenerative Markov chains
Patrice Bertail,Gabriela Ciolek +1 more
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In this paper, the Bernstein and Hoeffding type inequalities for regenerative Markov chains were presented and generalized to the case of Markov chain, and exponential bounds for suprema of empirical processes over a class of functions whose size is controlled by its uniform entropy number were established.Abstract:
The purpose of this paper is to present Bernstein and Hoeffding type inequalities for regenerative Markov chains. Furthermore, we
generalize these results and establish exponential bounds for suprema of empirical processes over a class of functions F which size is controlled by its
uniform entropy number. All constants involved in the bounds of the considered inequalities are given in an explicit form which can be advantageous
for practical considerations. We present the theory for regenerative Markov chains, however the inequalities are also valid in the Harris recurrent case.read more
Citations
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Journal ArticleDOI
Empirical Processes in M-Estimation
TL;DR: In this article, empirical processes in M-Estimation are studied. But they do not consider the effect of M-values on the accuracy of the M-estimation process.
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Concentration inequalities for Markov chains by Marton couplings and spectral methods
TL;DR: In this paper, a pseudo spectral gap is introduced for non-reversible Markov chains, which plays a similar role for nonreversible chains as the spectral gap plays for reversible chains.
Journal ArticleDOI
General Bernstein-Like Inequality for Additive Functionals of Markov Chains
TL;DR: In this article, the authors prove Bernstein-like inequalities for additive functionals of geometrically ergodic Markov chains, thus obtaining counterparts of inequalities for sums of independent random variables.
Journal ArticleDOI
Exponential inequalities for nonstationary Markov Chains
TL;DR: In this article, the authors extend the basic tools of Dedecker and Fan (2015) to nonstationary Markov chains and derive risk bounds for the prediction of periodic autoregressive processes with an unknown period.
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Mixing it up: A general framework for Markovian statistics beyond reversibility and the minimax paradigm
TL;DR: In this article, it was shown that mixing properties are sufficient to obtain deviation and moment bounds for integral functionals of general Markov processes, together with some unavoidable technical but not structural assumptions on the semigroup, which allow to derive convergence rates for kernel invariant density estimation which are known to be optimal in the much more restrictive context of reversible continuous diffusion processes.
References
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Book ChapterDOI
Probability Inequalities for sums of Bounded Random Variables
TL;DR: In this article, upper bounds for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt are derived for certain sums of dependent random variables such as U statistics.
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Markov Chains and Stochastic Stability
Sean P. Meyn,Richard L. Tweedie +1 more
TL;DR: This second edition reflects the same discipline and style that marked out the original and helped it to become a classic: proofs are rigorous and concise, the range of applications is broad and knowledgeable, and key ideas are accessible to practitioners with limited mathematical background.
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Weak Convergence and Empirical Processes: With Applications to Statistics
TL;DR: In this article, the authors define the Ball Sigma-Field and Measurability of Suprema and show that it is possible to achieve convergence almost surely and in probability.
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Weak Convergence and Empirical Processes
TL;DR: This chapter discusses Convergence: Weak, Almost Uniform, and in Probability, which focuses on the part of Convergence of the Donsker Property which is concerned with Uniformity and Metrization.
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Concentration Inequalities: A Nonasymptotic Theory of Independence
TL;DR: Deep connections with isoperimetric problems are revealed whilst special attention is paid to applications to the supremum of empirical processes.
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