Suppose $\mathcal{F} = \{f_x : x \in \mathbb{R}^d \}$ and each $f_x$ shares the same law $P$. If $\mathcal{F}$ is a class of uniformly bounded functions satisfying $L_r$-continuity, i.e. $\forall f \in \mathcal{F}$, we have

  1. $\| f\|_{\infty} : = \max_{x \in \mathbb{R}^d} |f(x)| < \infty$.

  2. $\mathrm{E} \sup_{|x - y| \leq \delta} |f_x - f_y|^r \leq \delta^{v}$,

then the article said the covering number of $\mathcal{F}$ satisfies $$N \big(\epsilon, \mathcal{F}, L_r(P) \big) \leq C \left( \frac{1}{\epsilon} \right)^{\frac{d}{v}}$$ for some positive constant $C >0$.

I am extremely confused about this result.

Could anyone help me? Thanks in advance.

  • $\begingroup$ I try to derive this result from the VC dimension, but the final result seems not related to the VC dimension of $\mathcal{F}$. $\endgroup$
    – 香结丁
    May 16, 2021 at 13:08
  • 1
    $\begingroup$ Can you let us know what the article is? $\endgroup$ May 16, 2021 at 14:27
  • $\begingroup$ @IosifPinelis In p34 of {L Su and HL White (2012), Conditional Independence Specification Testing for Dependent Processes with Local Polynomial Quantile Regression}. In their article $\tau$ is one-dimensional and the dimension of $\gamma$ is $d_{\Gamma}$, and they did not assume the boundedness of $(\tau, \gamma)$ (if $(\tau, \gamma)$ is in a bounded space the result of this covering number is direct). $\endgroup$
    – 香结丁
    May 17, 2021 at 3:10
  • $\begingroup$ This looks like something from Talagrand's 2014 book, Theorem B.3.3 $\endgroup$ Feb 5, 2022 at 16:17


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