Exponential Inequalities for Some Mixing Processes and Dynamic Systems
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929761903706112 |
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| author | Yuan, Zihao Dette, Holger |
| author_facet | Yuan, Zihao Dette, Holger |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_11481 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Exponential Inequalities for Some Mixing Processes and Dynamic Systems Yuan, Zihao Dette, Holger Statistics Theory Methodology 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. |
| title | Exponential Inequalities for Some Mixing Processes and Dynamic Systems |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2208.11481 |