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Exponential Inequalities for Some Mixing Processes and Dynamic Systems

Zihao Yuan, Holger Dette

TL;DR

This work introduces a unifying dependence framework, $\mathcal{C}_{p,\mathcal{F}}$-mixing, to cover non-strongly mixing dynamics and time series. It develops sharp exponential-type bounds for empirical means under both $p=\infty$ (equivalent to $\mathcal{C}_{\infty,\mathcal{F}}$-mixing) and $p=1$ mixing, including a refined Bernstein inequality and a milder-condition bound with a vanishing remainder. The results apply to general measurable spaces and do not require stationarity in the $p=\infty$ case, with a concrete application to a kernel-based estimator of the local conditional mode set under weak smoothness, achieving explicit Hausdorff-rate bounds. Overall, the framework subsumes classical mixing notions (e.g., $\alpha$-, $\tau$-, $\mathcal{C}$-mixing) and broadens the applicability to dependent data arising in time series and dynamical systems.

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.

Exponential Inequalities for Some Mixing Processes and Dynamic Systems

TL;DR

This work introduces a unifying dependence framework, -mixing, to cover non-strongly mixing dynamics and time series. It develops sharp exponential-type bounds for empirical means under both (equivalent to -mixing) and mixing, including a refined Bernstein inequality and a milder-condition bound with a vanishing remainder. The results apply to general measurable spaces and do not require stationarity in the case, with a concrete application to a kernel-based estimator of the local conditional mode set under weak smoothness, achieving explicit Hausdorff-rate bounds. Overall, the framework subsumes classical mixing notions (e.g., -, -, -mixing) and broadens the applicability to dependent data arising in time series and dynamical systems.

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 -mixing to cover such cases, where assumptions on the dependence structure become stronger with increasing We derive a series of sharp exponential-type (or Bernstein-type) inequalities under this dependence concept for and . More specifically, -mixing is equal to the widely discussed -mixing \citep{maume2006exponential}, and we prove a refinement of an Berntsein-type inequality in \cite{hang2017bernstein} for -mixing processes under more general assumptions. As there exist many stochastic processes and dynamic systems, which are not (or )-mixing, we derive Bernstein-type inequalities for -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.
Paper Structure (8 sections, 6 theorems, 124 equations)

This paper contains 8 sections, 6 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. $\mathcal{C}_{p,\mathcal{F}}$-mixing processes
  3. Exponential Inequalities
  4. $\mathcal{C}_{\infty,\mathcal{F}}$-mixing Processes
  5. $\mathcal{C}_{1,\mathcal{F}}$-mixing processes
  6. Conditional Mode Set Estimation
  7. Conclusions and Discussion
  8. Proof of Main Results

Key Result

Proposition 2.2

If the coefficient $C_p^*(k)$ of the $\mathbf{Z}$-valued stochastic process $(Z_{n})_{n\in\mathbb{N}}$ converges to $0$, $(Z_{n})_{n\in\mathbb{N}}$ is $\mathcal{C}_{p,\mathcal{F}}$-mixing, and for any $f\in\mathcal{F}$, the inequality holds, where ${1 \over p} + {1 \over q} =1$ and $C(k)=B^{-1}C_{p}^{*}(k)$.

Theorems & Definitions (9)

  • Definition 2.1
  • Proposition 2.2
  • Example 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.6
  • Remark 4.3
  • Proposition 4.5
  • Theorem 4.7