This paper gives upper and lower bounds for moments of sums of indepen
dent random variables (X-k) which satisfy the condition P(\X\(k) great
er than or equal to t) = exp(-N-k(t)), where N-k are concave functions
. As a consequence we obtain precise information about the tail probab
ilities of linear combinations of independent random variables for whi
ch N(t) = \t\(r) for some fixed 0 < r less than or equal to 1. This co
mplements work of Gluskin and Kwapien who have done the same for conve
x functions N.