A Bernstein type inequality and moderate deviations for weakly dependent sequences

Abstract

In this paper we present a 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 include classes of Markov chains, functions of linear processes with absolutely regular innovations and ARCH models.

Covariance selection by thresholding the sample correlation matrix

“…These three conditions are similar to the conditions in Theorem 1 of Merlevede et al (2011). Similar to Theorem 1, we have Theorem 3.…”

“…Under Conditions 3, 4 and 5, by Theorem 1 in Merlevede et al (2011) we can obtain a Bernstein type inequality for the sample correlation coefficients as in Proposition 1. The rest of the proof is similar to the proof of Theorem 1.…”

Risks of large portfolios

“…The stated strong-mixing condition is also standard in the literature (e.g., Merlevède et al (2011)). Note that it also follows from the a -mixing condition that ${\sum }_{h=1}^{\infty }\mid {\gamma }_T(h)\mid =Ofalse(1false)$ and ${\sum }_{h=1}^{\infty }\mid {\gamma }_f(h)\mid =Ofalse(1false)$ (see Lemma B.6 in the appendix) and hence Condition 4.4 holds easily as noted before.…”

High dimensional covariance matrix estimation using a factor model

“…By Assumption 3.3, r 4 < 1, and by the Bernstein inequality for weakly dependent data in Merlevède, Peligrad and Rio [(2009)…”