Previously, I asked a question here about the rate of convergence of expectations of absolute values to the expected value of a Gaussian.
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
$$\mathbb{E}\left|\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\right|\to\mathbb{E}|W|.$$
But uniform integrability guarantees the convergence $\mathbb{E}\left|\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\right|\to\mathbb{E}|W|.$ How fast does it converge?
My question today is: Do we have uniform integrability in this specific example for all moments? In other words, for what $p$ is the following true?
$$\mathbb{E}\left|\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\right|^p\to\mathbb{E}|W|^p.$$
I know the answer is YES for $p=1,2$ but suspect it is NO for $p=3,4,\ldots.$
We can prove that for each positive integer $p$, there exists a constant $C_p$ such that $\sup_n\mathbb E|S_n|^{2p}/n^p\leqslant C_p$, where $S_n=\sum\limits_{j=1}^nZ_i$. This can be done by induction on $p$, noticing that $$\mathbb E|S_n|^{2(p+1)}=\sum_{i=0}^{p+1}\binom{2p +2}{2i}\mathbb E|S_n|^{ 2i}.$$ It follows that for each positive real number $r$, the family $ \left\{\left(|S_n|/\sqrt n\right)^r, n\geqslant 1 \right\}$ is uniformly integrable.