Setup: Let $X $ be a random walk in discrete time, i.e.,
$$ X_t = X_{t-1}+ \epsilon_t $$
where the $\epsilon_t $ are i.i.d. $N(0, \sigma^2). $
Question. I am trying to use Donsker's Functional Central Limit Theorem (or Invariance Principle) to prove that
$$ n^{-5/2}\sum_{t=1}^n tX_{t-1} \stackrel{L}{\to} \int_0^1 rW(r)dr $$
where $W $ stands for Wiener process and $\stackrel{L}{\to} $ denotes convergence in law. The result is given in several papers starting in the late 1980s and in textbooks (it is a problem about time series models with unit roots), but nowhere it is more than stated. I am sort of convinced that this should be the result, but how can I make it rigorous?
The paper I am reading states it as if it were more or less obvious, so does even a textbook which then cites this paper where the result is not proved. I just cannot see how to completely justify the result. I understand that I can write it like this (and this probably helps):
$$ n^{-5/2}\sum_{t=1}^n tX_{t-1} = n^{-3/2}\sum_{t=1}^n \frac{t}{n}X_{t-1}. $$ However I cannot see how to write the last term as a function of $$ S_t^{(n)} = \frac{1}{\sigma\sqrt{n}}\sum_{i=1}^{\lfloor nt\rfloor}\epsilon_i \hskip 10pt 0 \le t \le 1 $$ (where $\lfloor a \rfloor $ denotes the largest integer less than or equal to $a $) which converges to $ W $ in law by Donsker's FCTL. Without the $t in the terms of the summation, I can manipulate these quantities to use the FCLT, but here, maybe I am missing something, but I cannot see how to do it in this case. I am starting to wonder if I even need to do this and maybe I am missing something really simple....
Could anyone please provide some help/guidance? Thank you.