Let $X_1,...,X_n$ i.i.d random variables, square integrable, and with $E[X_1]=0$.
Let $Y_n = \frac{|X_1 +...+X_n|}{\sqrt{n}}$
I am trying to show that $(Y_n)$ is uniformly integrable, i.e
$\sup_n\mathbb E[Y_n\mathbb 1\{Y_n\gt K\} ] \to 0$ when $K \to +\infty$
I have tried to use Cauchy-Schwarz inequality:
$\mathbb E[Y_n\mathbb 1\{Y_n\gt K\} ]^2 \leq \mathbb E[Y_n^2] \mathbb E[1\{Y_n\gt K\}]=\text{Var}(X_1) \mathbb P(Y_n>K)$
By the CLT, $\mathbb P(Y_n>K) \to \mathbb P(Y>K)$ where $Y$ follows $|\mathcal{N}(0,1)|$.
Also $\mathbb P(Y>K) \to 0$ when $K \to +\infty$.
Ideally I would like to have $\mathbb E[Y_n\mathbb 1\{Y_n\gt K\} ] \leq \epsilon(K) $ where $\epsilon(K) \to 0$ but I don't see how to get it.
Thanks for your help.