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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 $\theta$.

QED
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  • Welcome to Math.SE! Please, consider updating your question to include what you have tried / where you are getting stuck. You will find that people on this site will be significantly faster to help you if you do that; that way, we know exactly what help you need. – Did Aug 27 '13 at 08:04
  • I have done several problems where I had to find the scaling constants $a_n$ and $b_n$ but I am clueless, about how to find a function $h$ suitable for the problem. Initially I tried with Delta method but that took me nowhere. – QED Aug 27 '13 at 10:01

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

Did
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