In a paper from Nickl I found a theorem (Theorem 4) with a form of central limit theorem $$\sqrt{n}(P_n-P)\rightarrow G$$ in $l^\infty(F)$ where $P$ is a law on $\mathbb{R}$, $F$ is a class of function and $G$ is a gaussian process indexed by $f\in F$. But the central limit theorem which I'm familiar with is of the form: $$f_n-f\rightarrow G$$ where $G$ in indexed by $x\in\mathbb{R}$. I am looking for help to understand the first version and also would like to know if one can transform it to the second version? Is it correct to demand $f\in F$ in the second version? Thanks a lot for any hint!

In a paper from Nickl I found a theorem (Theorem 4) with a form of central limit theorem $$\sqrt{n}(P_n-P)\rightarrow G$$ in $l^\infty(F)$ where $P$ is a law on $\mathbb{R}$, $F$ is a class of function and $G$ is a gaussian process indexed by $f\in F$. But the central limit theorem which I'm familiar with is of the form: $$\sqrt{n}(f_n-f)\rightarrow G$$ where $G$ in indexed by $x\in\mathbb{R}$. I am looking for help to understand the first version and also would like to know if one can transform it to the second version? Is it correct to demand $f\in F$ in the second version? Thanks a lot for any hint!

In a paper from Nickl I found a theorem 4 with a form of central limit theorem $$\sqrt{n}(P_n-P)\rightarrow G$$ in $l^\infty(F)$ where $P$ is a law on $\mathbb{R}$, $F$ is a class of function and G a gaussian process index by $f\in F$. But the central limit theorem which I'm familiar with is like $$f_n-f\rightarrow G$$ where G in index by $x\in\mathbb{R}$. So how to understand the first version and can one transform it to the second version? Is it correct to demand $f\in F$ in the second version? Thanks a lot for any hint!

In a paper from Nickl I found a theorem (Theorem 4) with a form of central limit theorem $$\sqrt{n}(P_n-P)\rightarrow G$$ in $l^\infty(F)$ where $P$ is a law on $\mathbb{R}$, $F$ is a class of function and $G$ is a gaussian process indexed by $f\in F$. But the central limit theorem which I'm familiar with is of the form: $$f_n-f\rightarrow G$$ where $G$ in indexed by $x\in\mathbb{R}$. I am looking for help to understand the first version and also would like to know if one can transform it to the second version? Is it correct to demand $f\in F$ in the second version? Thanks a lot for any hint!