# A result of the covering number

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 try to derive this result from the VC dimension, but the final result seems not related to the VC dimension of $\mathcal{F}$.
• @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).