Let $X_1,...,X_n$ i.i.d random variables, square integrable, and with $E[X_1]=0$.
I am trying to find a simple asymptotic equivalent when $n \to +\infty$ of :
$E[|X_1+...+X_n|]$
I think the result is $\sqrt{\frac{2n}{\pi}E[X_1^2]}$ (derived easily when $X_i$ are $\mathcal{N}(0,\sigma^2)$ )
Furthermore, if I use the CLT, I get the same result assuming I can pass the limit into the expectation (which is a big assumption).
Thanks for your help !
We indeed have $|X_1+\dots+X_n|/\sqrt n\to |N|$ in distribution where $N$ is Gaussian, centered and $\mathbb EN^2=\mathbb E[X_1^2]$.
To conclude the convergence of the expectations, we need uniform integrability. Indeed, if $(Y_n)_{n\geqslant 1} $ is a uniformly integrable sequence of non-negative random variables which converges in distribution to $Y$, then $\mathbb E[Y_n] \to\mathbb E[Y]$. To see this, note that $$|\mathbb E[Y_n]-\mathbb E[Y]|\leqslant\int_0^R |\mathbb P\{Y_n\gt t \}-\mathbb P\{Y\gt t \}|\mathrm dt + \int_R^{\infty}\mathbb P\{Y_n\gt t \}\mathrm dt +\int_R^{\infty}\mathbb P\{Y\gt t \}\mathrm dt.$$ The second term is controlled by $\mathbb E[Y_n\mathbb 1\{Y_n\gt R\} ]$, hence using uniform integrability we can make this term arbitrarily small and uniformly in $n$. To handle the first term, use pointwise convergence of the cumulative distribution functions.