A tail inequality for suprema of unbounded empirical processes with applications to Markov chains

By Radosław Adamczak. 2007

229 citations recorded by the source corpus.

We present a tail inequality for suprema of empirical processes generated by variables with finite $\psi_\alpha$ norms and apply it to some geometrically ergodic Markov chains to derive similar estimates for empirical processes of such chains, generated by bounded functions. We also obtain a bounded difference inequality for symmetric statistics of such Markov chains.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2b12069aa90204725ec65410ad37984569ddc34c

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In Applied Mathematics

In Statistics and Probability

Citations in the corpus

Nearly optimal central limit theorem and bootstrap approximations in high dimensions (2020, background): “j,k,l,q=1 IE[U(1− U)ξijξikξilξiq∂jklqh(W (s) + √ 1− sUU ξi)], and U ′ is a uniform random variable on [0, 1] independent of everything else.”
A Bernstein type inequality and moderate deviations for weakly dependent sequences (2009, background): “0 P ( 2Γn−1 > [C/u] ) (1− u)n−1du := I((1)) n (x, y) + I (2) n (x, y) (4.”
Concentration inequalities for Markov chains by Marton couplings and spectral methods (2012, background): “Note that there are similar truncation arguments in the literature for ergodic averages of unbounded functions of Markov chains, see [1], [2], and [38].”
Bernstein inequality and moderate deviations under strong mixing conditions (2012, background): “In the context of bounded functions f of stationary geometrically strongly mixing Markov chains, Theorem 6 in Adamczak [1] provides a Bernstein type inequality for Sn(f) = f(X1) + · · · + f(Xn) with the factor log n instead of (log n)(2), which appears in (2.”
Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression (2018, methodology): “There are two differences of Theorem S.1.1 in comparison with Theorem 4 of Adamczak (2008).”
Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications (2016, unknown): “there exists a constant C(α) > 0 such that ‖ξ1 + ξ2‖ψα ≤ C(α)(‖ξ1‖ψα + ‖ξ2‖ψα) holds for all ξ1, ξ2 ∈ Lψα [1].”
A Bernstein–von Mises theorem for smooth functionals in semiparametric models (2013, methodology): “For any φ in L(f0), let us write F0(φ) as shorthand for ∫ 1 0 φ(x)f0(x)dx and set, for any positive density f on [0, 1], η = log f, η0 = log f0, h = √ n(η − η0).”
Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method (2018, methodology): “In order to apply the bounded version, we want to truncate each variable Xi, which can be done by the approach from Adamczak (2008) (see reference therein for more details on the applications of this method), where it was used in the context of Talagrand’s concentration inequality.”
The Bernstein–Orlicz norm and deviation inequalities (2011, background): “occurring in [1] by a constant proportional to K , which means we generally gain a log n-term.”
A Swiss Army Infinitesimal Jackknife (2018, background): “bound on a centered empirical process of derivatives of the objective function. Controlling quantities like these for unbounded g n() and h n() is an active topic in theoretical statistics (e.g. 8 [Adamczak, 2008, Mendelson, 2010]), though applying these results will be beyond the scope of the prese”
Finite sample change point inference and identification for high‐dimensional mean vectors (2017, unknown): “, there exists a constant C(β) > 0 such that ‖X + Y ‖ψβ 6 C(β)(‖X‖ψβ + ‖Y ‖ψβ) holds for all X,Y ∈ Lψβ (Adamczak, 2008).”