Assume $(X_i)_{i=1}^{\infty}$ is a sequence of i.i.d. random variables such that $\mathbb E[X_1] = 0$ and $\mathrm{Var}(X_1) = \sigma^2$. Define the partial sum $S_n(t)$ as
\begin{align} S_n(t) =\left\{\begin{array}{ll}& \frac{1}{n^{\alpha/2}}\sum_{i=n}^{n+tn^{\alpha}}X_i, &\mathrm{for}~t\geq0\\ & \frac{1}{n^{\alpha/2}}\sum_{i=n+tn^{\alpha}}^{n}X_i, &\mathrm{for}~-n^{1-\alpha}\leq t<0. \end{array}\right. \end{align}
for some $\alpha\in(0,1)$. Does $S_n(t)$, as a function of $t$, converge in distribution to a function over $(-\infty,+\infty)$? If so, to which function?