  We derive a lower bound for moments of random chaoses of order two with
coefficients in arbitrary Banach space F generated by independent symmetric
random variables with logarithmically concave tails (which is probably
two-sided). We also provide two upper bounds for moments of such chaoses when F
= L_q. The first is true under the additional subgaussanity assumption. The
second one does not require additional assumptions but is not optimal in
general. Both upper bounds are sufficient for obtaining two-sided moment
estimates for chaoses with values in Lq generated by Weibull random variables
with shape parameter greater or equal to 1.
