A function $\widehat{f}(x) \rightarrow f(x)$ uniformly over a compact interval $[\underline{x},\overline{x}]$, in addition, for some sequence ${a_{n}}$, $\forall x\in [\underline{x},\overline{x}]$, $a_{n}(\widehat{f}(x)-f(x))\rightarrow_{d} N(0,V(x))$ for some $V(x)$.
Is it true that $a_{n}(\widehat{f}(x)-f(x))\rightarrow G$ over $[\underline{x},\overline{x}]$, where $G$ is a mean-zero Gaussian process?
Thank you.
