Transforming a sequence of i.i.d. variables so that its asymptotic distribution is non-degenrate

Question

Suppose $X_1,X_2,\cdots$ are i.i.d. $U(0,\theta)$ random variables. Can you suggest a function $h$ of $X_1,\cdots,X_n$ and constants $a_n$ and $b_n$ such that $a_n(h(X_1,\cdots,X_n)-b_n)\xrightarrow{d}Y$ where $Y$ is a non-degenerate random variable whose distribution is independent of $\theta$.

Answer 1

$$\sqrt{n}\,\left(\frac{X_1+\cdots+X_n}{n\theta}-\frac12\right)$$