Uniform Chernoff and Dvoretzky-Kiefer-Wolfowitz-Type Inequalities for Markov Chains and Related Processes

Kontorovich, Aryeh; Weiss, Roi

doi:10.1239/jap/1421763330

Cited by 20 publications

(33 citation statements)

References 32 publications

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“…Note that a similar bound was obtained in Kontorovich and Weiss (2012). The main advantage of Proposition 2.19 is that the constants in the exponent of our inequality are proportional to the mixing time of the chain.…”

Section: Applicationssupporting

confidence: 73%

Concentration inequalities for Markov chains by Marton couplings and spectral methods

Paulin

2015

Electron. J. Probab.

167

193

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We prove a version of McDiarmid's bounded differences inequality for Markov chains, with constants proportional to the mixing time of the chain. We also show variance bounds and Bernstein-type inequalities for empirical averages of Markov chains. In the case of non-reversible chains, we introduce a new quantity called the "pseudo spectral gap", and show that it plays a similar role for non-reversible chains as the spectral gap plays for reversible chains.Our techniques for proving these results are based on a coupling construction of Katalin Marton, and on spectral techniques due to Pascal Lezaud. The pseudo spectral gap generalises the multiplicative reversiblication approach of Jim Fill.

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Section: Applicationssupporting

confidence: 73%

Concentration inequalities for Markov chains by Marton couplings and spectral methods

Paulin

2015

Electron. J. Probab.

167

193

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“…Now, taking g = I G we will arrive at (6.6). In fact, in our circumstances we can obtain (6.6) directly from [21] and [20]. Now we conclude the proof of (6.5) and of the whole Lemma 4.1 (in the setup of Theorem 2.7) in the same way as in the previous section.…”

Section: Finite State Space Casesupporting

confidence: 74%

A Nonconventional Local Limit Theorem

Hafouta

Kifer

2015

J Theor Probab

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Abstract. Local limit theorems have their origin in the classical De MoivreLaplace theorem and they study the asymptotic behavior as N → ∞ of probabilities having the form P {S N = k} where S N = N n=1 F (ξn) is a sum of an integer valued function F taken on i.i.d. or Markov dependent sequence of random variables {ξ j }. Corresponding results for lattice valued and general functions F were obtained, as well. We extend here this type of results to nonconventional sums of the form S N = N n=1 F (ξn, ξ 2n , ..., ξ ℓn ) which continues the recent line of research studying various limit theorems for such expressions.

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“…where we have used the trivial bound E π f 2 ≤ 1 to simplify the inequalities. Note that this yields an improvement over d 2 for j log d. Moreover, the bound (19) can itself be improved, since each f j is orthogonal to all eigenfunctions other than 1 and g j , so that the log d factors can all be removed by a more carefully argued form of Lemma 1. It thus follows directly from the bound (18) that if we draw N +T f j 2 samples, we obtain the tail bound…”

Section: Example: Lazy Random Walk On C 2dmentioning

confidence: 98%

“…The requirement that the chain start in equilibrium can be relaxed by adding a correction for the burn-in time [27]. Extensions of this and related bounds, including bounded-differences-type inequalities and generalizations to continuous Markov chains and non-Markov mixing processes have also appeared in the literature (e.g., [19,29]). The concentration result has an alternative formulation in terms of the mixing time instead of the spectral gap [4].…”

Section: Related Workmentioning

confidence: 99%

Function-Specific Mixing Times and Concentration Away from Equilibrium

Rabinovich¹,

Ramdas²,

Jordan³

et al. 2020

Bayesian Anal.

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Slow mixing is the central hurdle when working with Markov chains, especially those used for Monte Carlo approximations (MCMC). In many applications, it is only of interest to estimate the stationary expectations of a small set of functions, and so the usual definition of mixing based on total variation convergence may be too conservative. Accordingly, we introduce function-specific analogs of mixing times and spectral gaps, and use them to prove Hoeffding-like function-specific concentration inequalities. These results show that it is possible for empirical expectations of functions to concentrate long before the underlying chain has mixed in the classical sense, and we show that the concentration rates we achieve are optimal up to constants. We use our techniques to derive confidence intervals that are sharper than those implied by both classical Markov chain Hoeffding bounds and Berry-Esseen-corrected CLT bounds. For applications that require testing, rather than point estimation, we show similar improvements over recent sequential testing results for MCMC. We conclude by applying our framework to real data examples of MCMC, providing evidence that our theory is both accurate and relevant to practice.

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Uniform Chernoff and Dvoretzky-Kiefer-Wolfowitz-Type Inequalities for Markov Chains and Related Processes

Cited by 20 publications

References 32 publications

Concentration inequalities for Markov chains by Marton couplings and spectral methods

Concentration inequalities for Markov chains by Marton couplings and spectral methods

A Nonconventional Local Limit Theorem

Function-Specific Mixing Times and Concentration Away from Equilibrium

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