Exponential Inequalities for Some Mixing Processes and Dynamic Systems

TL;DR

This paper introduces a new dependence concept called $\\mathcal{C}_{p,\mathcal{F}}$-mixing for dynamic systems and time series, deriving sharp exponential inequalities applicable to a broad class of non-strong mixing processes.

Contribution

It generalizes existing mixing conditions by defining $\\mathcal{C}_{p,\mathcal{F}}$-mixing and establishes exponential inequalities for these processes, including cases where traditional mixing assumptions do not hold.

Findings

Derived sharp Bernstein-type inequalities for $\\mathcal{C}_{p,\mathcal{F}}$-mixing processes.

Refined inequalities for $\\mathcal{C}$-mixing processes under broader assumptions.

Applied inequalities to analyze convergence rates of estimators in less smooth density scenarios.

Abstract

Many important dynamic systems, time series models or even algorithms exhibit non-strong mixing properties. In this paper, we introduce the general concept of $\mathcal{C}_{p,\mathcal{F}}$-mixing to cover such cases, where assumptions on the dependence structure become stronger with increasing $p\in [1, \infty].$ We derive a series of sharp exponential-type (or Bernstein-type) inequalities under this dependence concept for $p=1$ and $p=\infty$. More specifically, $\mathcal{C}_{\infty,\mathcal{F}}$-mixing is equal to the widely discussed $\mathcal{C}$-mixing \citep{maume2006exponential}, and we prove a refinement of an Berntsein-type inequality in \cite{hang2017bernstein} for $\mathcal{C}$-mixing processes under more general assumptions. As there exist many stochastic processes and dynamic systems, which are not $\mathcal{C}$ (or $\mathcal{C}_{\infty,\mathcal{F}}$)-mixing, we derive Bernstein-type inequalities for $\mathcal{C}_{1,\mathcal{F}}$-mixing processes as well and we use this result to investigate the convergence rates of plug-in-type estimators of the local conditional mode set for vector-valued output, in particular in situations where the density is less smooth.