I recently asked a question here that was the following:
If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = P(Z_i=+1) = \frac 12,$ then we have by the Central Limit Theorem that $\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\stackrel{d}{\to} \mathcal{N}(0,1),$ so that for any continuous bounded function $f,$ we have $\mathbb{E}f\left(\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\right)\to\mathbb{E}f(W)$ where $W\sim\mathcal{N}(0,1).$ Now, $|\cdot|$ is not a bounded function, so it is not necessarily true that
$$\mathbb{E}\left|\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\right|\to\mathbb{E}|W|.$$
My question is whether the above is true for this specific distribution of $Z_i.$ If not, what does $\mathbb{E}\left|\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\right|$ converge to (if anything)?
I learnt from the answer given that uniform integrability guarantees the convergence $\mathbb{E}\left|\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\right|\to\mathbb{E}|W|.$ However, I am interested in knowing the rate of convergence for this particular example. Can you help me with some bound on how fast the sequence converges?
Define $Y_n:=n^{-1/2}|\sum_{j=1}^nZ_j|$.
From the Berry-Esseen theorem, we have for some universal constant $C$, $$\tag{1}|\mu\{Y_n>t\}-\mu\{|W|>t\}|\leqslant \frac C{\sqrt n}.$$ For any positive $R$, $\mathbb E[Y_n\chi_{\{Y_n>R\}}|\leqslant R^{-1}$ and a similar inequality holds for $|W|$. Using (1) and the formula $\mathbb E[X]=\int_0^{+\infty}\mu\{X>t\}\mathrm dt$ valid for a non-negative random variable $X$, we obtain $$\tag{2}|\mathbb E[Y_n]-\mathbb E|Z||\leqslant C\left(\frac{R}{\sqrt n}+\frac 1R\right).$$ Optimizing the RHS of (2) in $R$, we obtain that for some constant $C$ independent of $n$, $$|\mathbb E[Y_n]-\mathbb E|Z||\leqslant \frac{C}{n^{1/4}}.$$