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The Glivenko-Cantelli Theorem (http://en.wikipedia.org/wiki/Glivenko%E2%80%93Cantelli_theorem) states that if $F$ is a distribution function, $X_1,\dots,X_n \sim F$, and $\hat{F}_n$ is the empirical distribution function, then $$\sup_{x \in \mathbb{R}} \lvert \hat{F}_n(x) - F(x) \rvert \xrightarrow{a.s.} 0 . \tag{1}$$

How does this differ from simply stating the following? $$\hat{F}_n(x) \xrightarrow{a.s.} F(x) \tag{2}$$

Using the definition of convergence almost surely, from (1): $$\begin{align*} \mathbb{P} \left( \lim_{n\rightarrow\infty} \left\lvert \sup_{x \in \mathbb{R}} \lvert \hat{F}_n(x) - F(x) \rvert \right\rvert = 0 \right) &= 1 \\ \mathbb{P} \left( \lim_{n\rightarrow\infty} \sup_{x \in \mathbb{R}} \lvert \hat{F}_n(x) - F(x) \rvert = 0 \right) &= 1 \tag{1a} \end{align*}$$

Using the definition of convergence almost surely, from (2): $$\begin{align*} \mathbb{P} \left( \lim_{n\rightarrow\infty} \left\lvert \hat{F}_n(x) - F(x) \right\rvert = 0 \right) = 1 \tag{2a} \end{align*}$$

To me, it seems that (1a) and (2a) are equivalent statements because of the least upper bound, and thus (1) and (2) are equivalent statements. But I have a feeling that I'm missing a subtle difference since otherwise I would think the Theorem would just be stated the simpler way (Equation (2)).

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The differerence is in uniform convergence. 1 is saying that for all x there is a single n such that error is less than epsilon ( uniform). The other one is saying for each x there is a large enough n that error is less than epsilon(pointwise). An example from wikipedia of pointwise but not uniform convergence is $ f_n (x)=x^n $ on $0\le x\le 1$. You need larger and larger n as you get closer to $x=1$.

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    $\begingroup$ Hmm. Did you really mean for that to end with "closer to $x=0$"? I'd have thought that the need for larger $n$ was near $x=1$. $\endgroup$ – Glen_b Sep 22 '14 at 22:04

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