Suppose that we have an iid sample $X_1,\dots,X_n$ with a distribution function $F$. Denote $\bar X_n:=\frac{1}{n}\sum_{i=1}^n X_i$ and $\bar X_n^*:=\frac{1}{n}\sum_{i=1}^n X_i^*,$ where $X_1^*,\dots,X_n^*$ are iid from the empirical distribution function $\hat F$ given the sample $X_1,\dots,X_n$. Does it then follow that $$\sup_x|\mathbb P(\sqrt{n}\bar X_n \leq x) - \mathbb P(\sqrt{n}\bar X_n^* \leq x)| \rightarrow 0 \ \ \mathrm{a.s.}$$
Approximation of sample mean distribution
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statistics
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1Shouldn't there be a limit $n\to \infty$ added? – 2012-03-06