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 $X$? .
convergence in distribution and convergence of quantile
4
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probability-theory
convergence
1 Answers
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Yes. If $X$ is a random variable with distribution function $F$, then for $0
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.
– 2011-12-25