Prove Rate of Convergence of Monte Carlo

Question

Let $X_1, X_2, \ldots$ be i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. How does

\begin{equation} \mathbb E\left[\,\left|\frac{1}{N} \sum_{i=1}^n X_i - \mu\, \right|\,\right] \to O\left(\frac{1}{\sqrt N}\right) \end{equation}

follow from the central limit theorem? We easily get

\begin{equation} \mathbb E\left[\left(\frac{1}{N} \sum_{i=1}^n X_i - \mu\, \right)^2\right] = \frac{\sigma^2}{N}, \end{equation}

but how to get the first one?

EDIT: Actually, any proof would do $-$ does not have to use the central limit theorem.

Answer 1

Since $r\mapsto(\mathsf{E}|\cdot|^r)^{1/r}$ is nondecreasing,

$$ \mathsf{E}|\bar{X}_N-\mu|\le \frac{\sigma}{\sqrt{N}}. $$

Answer 2

Use Jensen's inequality: $\phi(\mathbb{E}[Y])\le \mathbb{E}[\phi(Y)]$ for any convex function $\phi$. Take $Y=|\overline{X}_N-\mu|$ and $\phi(x)=x^2$.