This question is a repost from Mathematics Stack Exchange, where it did not receive any answer.

Assume $(X_i)_{i=1}^{\infty}$ is a sequence of i.i.d. real-valued random variables such that $\mathbb E[X_i^2]<\infty$. Denote by $F_X(t) := \mathbb P(X\leq t)$ their common distribution function. The regular Donsker's Theorem states that

\begin{equation}\tag{1} n^{1/2}\left(\frac{\sum_{i=1}^n \mathbb 1_{\{X_i\leq t\}}}{n}-F_X(t)\right)\stackrel{\mathrm d}{\rightarrow} B_0(F(t)), \end{equation}

where $\stackrel{\mathrm d}{\rightarrow}$ denotes convergence in distribution, and $B_0(\cdot)$ is a Brownian Bridge on $[0,1]$. My question is related to a previous one, which dealt with a generalization of the above result.

Let us replace $t$ by $\bar t + t n^{-\alpha}$ in $(1)$, for some fixed $\bar t >0$ and $0<\alpha <1$. Accordingly, we replace the scaling constant $n^{1/2}$ by $n^{(1+\alpha)/2}$. We obtain

\begin{equation}\tag{2} n^{(1+\alpha)/2}\left(\frac{\sum_{i=1}^n \mathbb 1_{\{X_i\leq \bar t + t n^{-\alpha}\}}}{n}-F_X(\bar t + t n^{-\alpha})\right). \end{equation}

My question is: does $(2)$ converge in distribution to some process (necessarily defined on $(-\infty,\infty)$)? If so, is there an explicit characterization? Has this been studied before?