Setting:
Say we have a sequence of random variables $\{X_i\}_{i=1...n}$ such that for $T_n(s)=n^{\beta}\sum_{i=1}^{[sn]}X_i$ we have $$T_n(s)\Rightarrow W(s)$$ as $n\to\infty$.
Question:
Is there an result as to how does the following limit looks like $$n^{\alpha}\left(T_n(s)-W(s)\right)$$ as $n\to\infty$? (Note that I would essentially need an $O_p(1)$ limit.)
Thanks for your comments and/or references in advance :-)