Tail and moment estimates for some types of chaos

Let ${\displaystyle X_i}$ be a sequence of independent symmetric real random variables with logarithmically concave tails. We consider a variable ${\displaystyle X={\sum }_{i\ne j}{a}_{i,j}{X}_i{X}_j}$, where ${\displaystyle a_{i,j}}$ are real numbers. We derive approximate formulas for the tails and moments of X and of its decoupled version, which are exact up to some universal constants.