Suppose $X_1,X_2,...$ are bounded random variables with compact support, and $\frac{X_1+...+X_n}{\sqrt{n}}\overset{d}{\longrightarrow}N(0,1)$.
Is there neccessarily a central limit theorem for the sequence of random variables $Y_n=\frac{X_n}{\sqrt{n}}$ just with a different scaling ($\sqrt{\log n}$ instead of $\sqrt{n}$)?