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

Let ${\displaystyle X_i}$ be a sequence of independent symmetric real random variables with logarithmically concave tails. We consider a variable ${\displaystyle X=\sum v_i{X}_i}$, where ${\displaystyle v_i}$ are vectors of some Banach space. We derive approximate formulas for the tail and moments of ∥X∥. The estimates are exact up to some universal constant and they extend results of S. J. Dilworth and S. J. Montgomery-Smith [1] for the Rademacher sequence and E. D. Gluskin and S. Kwapień [2] for real coefficients.