A Bernstein type inequality and moderate deviations for weakly dependent sequences

By F. Merlevède, Magda Peligrad, E. Rio. 2009

210 citations recorded by the source corpus.

In this paper we present a Bernstein-type tail inequality for the maximum of partial sums of a weakly dependent sequence of random variables that is not necessarily bounded. The class considered includes geometrically and subgeometrically strongly mixing sequences. The result is then used to derive asymptotic moderate deviation results. Applications are given for classes of Markov chains, iterated Lipschitz models and functions of linear processes with absolutely regular innovations.

Subfields: Management Science and Operations Research, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/66eedf6a66af780b1750b8fb6cbe6338aff1515d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Management Science and Operations Research

In Numerical Analysis

In Statistics and Probability

Citations in the corpus

SHRINKAGE ESTIMATION OF REGRESSION MODELS WITH MULTIPLE STRUCTURAL CHANGES (2015, background): “As Perron (2006) remarks, early works were mostly designed for the specific case of a single change.”
Hoeffding’s inequality for supermartingales (2011, methodology): “k ≤ v2 for some k ∈ [1,n]) ≤ exp ˆ − x2 2(v2 + 1 3 x2−α) ˙ +nC1exp{−δxα}. (31) For weakly dependent sequences, an inequality similar to (31) has recently been obtained by Merlev`ede, Peligrad and Rio [30]. For martingale diﬀerences ξi (i = 1,...,n) satisfying Cram´er’s condition Ee|ξi| ≤ C 1 for som”
On the rate of convergence in Wasserstein distance of the empirical measure (2013, unknown): “(b | | ∞ | | ≥ ≤ [31, Formula (1.4)] which is based on results by Borovkov [8].”
Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression (2018, background): “Observe that the RE condition is only concerned with the minimum sparse eigenvalue and so the assumption of exponential tails may not be required in its full strength; see van de Geer and Muro (2014) and Oliveira (2013) for details.”
Breaking the curse of dimensionality in regression (2017, methodology): “It follows by Theorem 1 of Merlevède et al. (2011) and the union bound that ∀c > 0 P ( max 1≤j≤d n−1 ∣∣∣∣∣ n∑ i=1 ζi,j ∣∣∣∣∣ ≥ c ) → 0.”
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].”
HIGH DIMENSIONAL COVARIANCE MATRIX ESTIMATION IN APPROXIMATE FACTOR MODELS (2011, unknown): “Hence again we can apply the Bernstein inequality for weakly dependent data in Mer-lev`ede, Peligrad and Rio [(2009), Theorem 1] and the Bonferroni method on Z ′ ki,t similar to (B.1) with the parameter γ and the rest terms on the right-hand side of the inequality, multiplied by pK are of order o (…”