  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. 
Moments and tails of Lq-valued chaoses based on independent variables   with log-concave tails