Let $X_0, X_1, X_2, \dots$ be a sequence of i.i.d. real-valued random variables and define the sample mean $S_n\equiv \frac 1n\sum_{i=0}^n X_i$. Suppose $\mathbb E (X_0)=0$ and $\text{Var}(X_0)=\sigma^2<+\infty$. Then a version of the Donsker's theorem states that for all $t\in [0,1]$ $$\frac{S_{\lfloor nt\rfloor}}{\sqrt n}\Rightarrow W(t)$$ on $C[0,1]$.
I am curious about whether the following statement is true (which seems like an infinite-dimensional version of the statement above):
With the same assumptions, for each fixed $t'\in [0,n]$, let $W_n(t')$ be the linear interpolation between the points $$\bigg(\lfloor nt\rfloor-1,\frac{S_{\lfloor nt\rfloor-1}}{\sqrt n}\bigg)\,\,\,\text{and }\,\,\,\bigg(\lfloor nt\rfloor, \frac{S_{\lfloor nt\rfloor}}{\sqrt n}\bigg)$$ where $t'\in (\lfloor nt\rfloor-1, \lfloor nt\rfloor]$. Then $W_n(t')\Rightarrow W(t')$ on $C[0,+\infty)$. What about if we use some smooth interpolation instead of the linear?
Can anyone give me a hint or some references? Many thanks!