A deviation bound for $\alpha$-dependent sequences with applications to intermittent maps

TL;DR

This paper establishes a deviation bound for sums of functions of $oldsymbol{ ext{alpha}}$-dependent sequences, extending existing inequalities and applying them to intermittent maps that are not $oldsymbol{ ext{alpha}}$-mixing, with implications for large deviations and invariance principles.

Contribution

It introduces a deviation bound for $oldsymbol{ ext{alpha}}$-dependent sequences and extends the Rosenthal inequality to this broader class, with applications to intermittent maps.

Findings

Derived a deviation inequality for $oldsymbol{ ext{alpha}}$-dependent sequences.

Extended Rosenthal inequality to $oldsymbol{ ext{alpha}}$-dependent sequences.

Applied results to intermittent maps, demonstrating non-$oldsymbol{ ext{alpha}}$-mixing behavior.

Abstract

We prove a deviation bound for the maximum of partial sums of functions of $\alpha$-dependent sequences as defined in Dedecker, Gou{\"e}zel and Merlev{\`e}de (2010). As a consequence, we extend the Rosenthal inequality of Rio (2000) for $\alpha$-mixing sequences in the sense of Rosenblatt (1956) to the larger class of $\alpha$-dependent sequences. Starting from the deviation inequality, we obtain upper bounds for large deviations and an H{\"o}lderian invariance principle for the Donsker line. We illustrate our results through the example of intermittent maps of the interval, which are not $\alpha$-mixing in the sense of Rosenblatt.