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.

翻译：许多重要的动力系统、时间序列模型乃至算法都展现出非强混合特性。本文引入 $\mathcal{C}_{p,\mathcal{F}}$-混合的一般概念以涵盖此类情形，其中对相依结构的假设随 $p\in [1, \infty]$ 增大而增强。我们针对 $p=1$ 和 $p=\infty$ 的情况，在此相依概念下推导了一系列尖锐的指数型（或伯恩斯坦型）不等式。具体而言，$\mathcal{C}_{\infty,\mathcal{F}}$-混合等价于被广泛讨论的 $\mathcal{C}$-混合 \citep{maume2006exponential}，我们在更一般的假设下证明了 \cite{hang2017bernstein} 中关于 $\mathcal{C}$-混合过程的伯恩斯坦型不等式的改进形式。由于存在许多并非 $\mathcal{C}$（或 $\mathcal{C}_{\infty,\mathcal{F}}$）-混合的随机过程与动力系统，我们也推导了 $\mathcal{C}_{1,\mathcal{F}}$-混合过程的伯恩斯坦型不等式，并利用该结果研究了向量值输出情形下局部条件众数集的插件型估计量的收敛速率，特别是在密度光滑性较差的场景中。