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Suppose real-valued random variables $\{X_{n}\} $ converges to $X$ in distribution. Then, will the quantile of the distribution of $\{X_n\}$ converge to the quantile of $X$? .

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Yes. If $X$ is a random variable with distribution function $F$, then for $0<p<1$ define the quantile function as $Q(p)=\inf(x: F(x)\geq p)$. Then $X_n\to X$ in distribution if and only if $Q_n(p)\to Q(p)$ at all continuity points $p$ of $Q$.

Added: It's a nice exercise to prove this result from the definition. On the other hand, it is Proposition 5 (page 250) in A Modern Approach to Probability Theory by Bert Fristedt and Lawrence Gray, and is also proved in Chapter 21 of Asymptotic Statistics by A. W. van der Vaart.

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But what if $Q$ is not continuous at $p$? –  webster Dec 25 '11 at 11:09
    
Suppose we know that $F$ is strictly increasing. Will this additional condition help? –  webster Dec 25 '11 at 14:08
    
The discontinuity points of $Q$ correspond to the "flat spots" on the graph of $F$ (Draw a picture!). So if $F$ is strictly increasing, then $Q$ is continuous at all $0<p<1$. –  Byron Schmuland Dec 25 '11 at 15:26
    
Yes. But what will be the formal proof? –  webster Dec 25 '11 at 16:25

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