论文标题
某些混合过程和动态系统的指数不平等现象
Exponential Inequalities for Some Mixing Processes and Dynamic Systems
论文作者
论文摘要
许多重要的动态系统，时间序列模型甚至算法都具有非弹性混合特性。 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 $和$ p = \ infty $。 更具体地说，$ \ MATHCAL {C} _ {\ INFTY，\ MATHCAL {F}} $ - 混合等于广泛讨论的$ \ Mathcal {C} $ - 混合\ citep {Maume2006 Expential}，并且我们对Berntseinein-typee insqu in equ in equ Inceporational in \ cite {hang2017bernstein}对于$ \ Mathcal {C} $ - 在更一般的假设下混合过程。由于存在许多随机过程和动态系统，这些过程不是$ \ Mathcal {c} $（或$ \ Mathcal {c} _ {\ infty，\ infty，\ Mathcal，\ Mathcal {f}} $） - 混合，我们得出$ \ natercal的bernstein-type type type type type type for $ \ nathcal {c} $ c} $}过程也是过程，我们使用此结果来研究为矢量值输出设置的局部条件模式的插件类型估计器的收敛速率，尤其是在密度较低的情况下。
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.
