Bernstein inequality and moderate deviations under strong mixing conditions

Abstract

In this paper we obtain a Bernstein type inequality for a class of weakly dependent and bounded random variables. The proofs lead to a moderate deviations principle for sums of bounded random variables with exponential decay of the strong mixing coefficients that complements the large deviation result obtained by Bryc and Dembo (1998) under superexponential mixing rates.

Square‐Root LASSO for High‐Dimensional Sparse Linear Systems with Weakly Dependent Errors

“…Repeating the argument as in the proof of Lemma , we get with where r ∈(0,1) is some constant to be chosen later. By , we get For I 1 , by the Bernstein inequality (Merlevede et al , ), we have …”

“…For I 1 , by the Bernstein inequality (Merlevede et al, 2009), we have where K > 0 only depends on the d in (2.2) and x ∶= (1−r) c . It is easy to see that the right-hand side of the above inequality is bounded by…”

A large deviation inequality forβ-mixing time series and its applications to the functional kernel regression model

“…In the special case that the aggregating function f (x, y) = g(x)h(y) is multiplicative, the statement follows from the large deviation inequalities for strong mixing processes given e.g. in Merlevède et al (2009). However, for a general aggregating function which does not allow a similar decomposition, we face the problem that the process given by Z k = f (X k , X t ) is not necessarily strong mixing any more.…”

Moment Bounds for Large Autocovariance Matrices Under Dependence

“…Hence by Lemma 3 in Merlevède et al (2009), for 0 < t ≤ { ψ(ψ 1 , ψ 2 , n, p)} −1 , we have log E Tr exp t n j=1 X j ≤ log p + t 2 15 √ nν + 60 1/ψ 2 2 1 − t ψ(ψ 1 , ψ 2 , n, p) .…”