$\epsilon_{t}$ is an IID(0,1) sequence. Let $k_{0}=[T\pi_{0}]$ for $\pi_{0}$ given/known and $\lambda \in [0,1]$. As $T\rightarrow \infty$, I would like to convince myself that the following two results hold as a standard application of the functional CLT.
$\displaystyle\frac{\sum_{t=k_{0}}^{k_{0}+[(T-k_{0})\lambda]}\epsilon_{t}}{\sqrt{T-k_{0}}} \Rightarrow W(\lambda)$
and
$\displaystyle\frac{\sum_{t=k_{0}}^{k_{0}+[(T-k_{0})\lambda]}\epsilon_{t}}{\sqrt{T}} \Rightarrow \sqrt{1-\pi_{0}} W(\lambda)$ (as I treat $\pi_{0}$ as given).
Thanks. I am not sure I got the top one right