# Probability Curiosity

Let $X_{1},X_{2},\ldots$ be i.i.d. random variables with $E\left[X_{1}\right]=0$ and $0<Var\left(X_{1}\right)=\sigma^{2}<\infty.$.

Let $S_{n}=\sum_{j=1}^{n}X_{n}$. Consider now $\lim_{n\to\infty}E\Big[ \frac{|{S_{n}}|}{\sigma\sqrt{n}}\Big]$

To which theorem I have to appeal in order to bring the limit inside?

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Inside of what? The expectation? – Sasha Mar 4 '13 at 12:48
@Sasha. Yes, inside expectation – Thinker Mar 4 '13 at 13:41

Let $T_n=|S_n|/(\sigma\sqrt{n})$, one knows that $T_n$ converges in distribution to the absolute value of a standard normal random variable hence, by Skorokhod's representation theorem, there exists some random variables $Z_n$ and $Z$ defined on a common probability space such that $Z$ is the absolute value of a standard normal, each $Z_n$ is distributed as $T_n$, and $Z_n\to Z$ almost surely.
Now, $(T_n)$ is bounded in $L^2$ hence $(Z_n)$ is bounded in $L^2$ hence $(Z_n)$ is uniformly integrable hence $Z_n\to Z$ in $L^1$, in particular $\mathbb E(T_n)=\mathbb E(Z_n)\to\mathbb E(Z)$.