TAIL AND MOMENT ESTIMATES FOR SUMS OF INDEPENDENT RANDOM-VARIABLES WITH LOGARITHMICALLY CONCAVE TAILS

For random variables S = Sigma(i=1)(infinity) alpha(i) xi(i), where (x i(i)) is a sequence of symmetric, independent, identically distributed random variables such that In P(\xi(i)\greater than or equal to t) is a concave function we give estimates from above and from below for th e tail and moments of S. The estimates are exact up to a constant depe nding only on the distribution of xi. They extend results of S. J. Mon tgomery-Smith [MS], M. Ledoux and M. Talagrand [LT, Chapter 4.1] and P . Hitczenko [H] for the Rademacher sequence.