Tail and moment estimates for a class of random chaoses of order two

We derive two-sided bounds for moments and tails of random quadratic forms (random chaoses of order $2$), generated by independent symmetric random variables such that $\parallel X{\parallel }_{2p}\le \alpha \parallel X{\parallel }_p$ for any $p\ge 1$ and some $\alpha \ge 1$. Estimates are deterministic and exact up to some multiplicative constants which depend only on $\alpha$.