Donsker's Theorem for triangular arrays I should mention that I already posed this question on Math Stack Exchange, but didn't receive much feedback.
Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $(\frac{X_i}{n^{\alpha}})_{i=1}^n$? More precisely, does
$$n^{2\alpha}\left(\frac{\sum_{i=1}^n \mathbf 1_{\{X_i\leq t n^{-\alpha}\}}}{n} - F_X(tn^{-\alpha})\right)\stackrel{\mathrm d}{\rightarrow} B(t),$$
where $B(\cdot)$ is a Brownian motion and $F_X$ is the distribution function of $X_1$, hold? I should mention that the $n^{2\alpha}$ is only an (informed) guess. 
In the usual Donsker's Theorem the limiting process is $B_0(F(t))$, where $B_0$ is a Brownian Bridge, but in this case the limiting process would need to be pinned down to zero 'at infinity', and thus my guess that it's actually a Brownian Motion.
Is this common knowledge? I have been digging through the literature and it does not seem to be proved anywhere.
 A: I guess you assume the $X_i$'s to take values in $[0,\infty)$. As it seems you are essentially rescaling in time as well, I would rather expect a convergence to a Poisson process.
Take for example $\alpha=1$. Then it is known that
$\sum_{i=1}^n \mathbf{1}_{\{X_i \leq tn^{-1}\}} \stackrel{d}{\longrightarrow} N(t),$
where $(N(t))_{t\geq 0}$ is a Poisson process with intensity function $f_X(0)t$ and  $f_X$ is the density function associated with $F_X$ (see for example Thm. 4.41 in [1]; for more details and stronger types of convergence see e.g. this paper).
Let's assume your your type of Donsker's theorem was correct. If we take $X$ to be exponentially distributed, say $F_X(x) = 1 - \exp(-x)$, then we can show via power series expansion $n^2F_X(t/n) = \mathcal{O}(n)$. But this would yield for $n \rightarrow \infty$
\begin{equation}
  \label{eq:2}
  n\Bigg(\underbrace{\sum_{i=1}^n \mathbf{1}_{\{X_i \leq tn^{-1}\}}}_{\rightarrow N(t)} + \,\mathcal{O}(1)\Bigg) \rightarrow +\infty.
\end{equation}
References
[1] Jacod, J. and A. N. Shiryaev (2003). Limit theorems for stochastic processes (Second ed.)
