Tail and moment estimates for sums of independent random variables with logarithmically concave tails

For random variables ${\displaystyle S={\sum }_{i=1}^{\infty }{\alpha }_i{\xi }_i}$, where ${\displaystyle \left({\xi }_i\right)}$ is a sequence of symmetric, independent, identically distributed random variables such that ${\displaystyle \ln P(\left|{\xi }_i\right|\ge t)}$ is a concave function we give estimates from above and from below for the tail and moments of S. The estimates are exact up to a constant depending only on the distribution of ξ. They extend results of S. J. Montgomery-Smith [MS], M. Ledoux and M. Talagrand [LT, Chapter 4.1] and P. Hitczenko [H] for the Rademacher sequence.