Suppose that independent and identically distributed random variable $X_1,...,X_n$ satisfying that expectation $EX_i=0$, for $i=1,...,n$, and for given value $r$ that $E|X_i|^r<\infty$, for $r>2$. How do I prove that the $E|\sum_{i=1}^n X_i|^r=O(n^{r/2})$ in a simple way？
I know that from this paper http://www.sciencedirect.com/science/article/pii/S0167715201000153, one can see that $E|\sum_{i=1}^n X_i|^r=O(n^{r/2})$. But is there a simple proof of this?
Any hint or proof would be appreciated.