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I am somewhat confused with the empirical distribution function:

Assume we have $X_1, X_2, ...$ iid real-valued random variables with true distribution $F_0$. Then by the Theorem of Glivenko-Cantelli,

$$| \hat F_n - F_0 |_\infty \longrightarrow 0$$

Does this also imply that $\hat F_n$ converges to the true CDF in $L^1$ and $L^2$ norm, i.e,

$$ \int | \hat F_n(x) - F_0(x) | \, dx \longrightarrow 0$$ and $$ \left[\int (\hat F_n(x) - F_0(x))^2 \, dx\right]^\frac{1}{2} \longrightarrow 0?$$

If not, what else doe we have to assume on $F_0$?

After reading on embeddings of $L^p$ spaces (http://en.wikipedia.org/wiki/Lp_space#Embeddings), I guess this does not follow from some general results, but is there maybe something special to the empirical CDF that guarantees this?

WLOG
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Emil
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  • According to this paper: http://www.ams.org/journals/tpms/2007-75-00/S0094-9000-08-00716-3/S0094-9000-08-00716-3.pdf

    it seems that a sufficient condition for $L^p$ convergence is that $X_1\in L^p$.

     – Math1000 Jan 09 '15 at 14:38
  • Thanks, that looks great! Does this by any chance generalize to dependent sequences in a simple way? – Emil Jan 09 '15 at 15:22

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