For the sum $S = \sum X_i$ of a sequence $(X_i)$ of independent symmetric (or nonnegative) random variables, we give lower and upper estimates of moments of $S$. The estimates are exact, up to some universal constants, and extend the previous results for particular types of variables $X_i$.
Rafał Latała. "Estimation of moments of sums of independent real random variables." Ann. Probab. 25 (3) 1502 - 1513, July 1997. https://doi.org/10.1214/aop/1024404522