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 $∥X∥_{2p}\leq α∥X∥_p$ for any $p≥1$ and some $α≥1$. Estimates are deterministic and exact up to some multiplicative constants which depend only on $α$.