I'm currently reading this paper: http://user.math.uzh.ch/barbour/pub/Barbour/SteinDiffusion.pdf and in equation (2.26) the author uses the following fact:
If $Z(t)$ is a standard Brownian Motion and $Z_n(t)=n^{-1/2}\sum_{k=1}^{\lfloor nt\rfloor}X_k$ for $X_k$ iid. standard normal then $Z(t)$ and $Z_n(t)$ can be realised together by first realising $Z(t)$ and then defining $Z_n(j/n)=Z(j/n)$, which then can be used to prove that:
$$\mathbb{E}\left[\sup_{t\in[0,1]}|Z_n(t)-Z(t)|\right]\leq Cn^{-1/2}\sqrt{\log n}$$ for some constant $C$.
Does anyone know how this last claim can actually be proved?