Moment inequalities for sums of certain independent symmetric random variables

This paper gives upper and lower bounds for moments of sums of independent random variables ${\displaystyle \left(X_k\right)}$ which satisfy the condition ${\displaystyle P({\left|X\right|}_k\ge t)=\exp (-N_k\left(t\right))}$, where ${\displaystyle N_k}$ are concave functions. As a consequence we obtain precise information about the tail probabilities of linear combinations of independent random variables for which ${\displaystyle N\left(t\right)={\left|t\right|}^r}$ for some fixed 0 < r ≤ 1. This complements work of Gluskin and Kwapień who have done the same for convex functions N.