  We obtain first decay rates of probabilities of tails of multivariate
polynomials built on independent random variables with heavy tails. Then we
derive stable limit theorems for nonconventional sums of the form $\sum_{Nt\geq
n\geq 1}F(X(q_1(n)),...,X(q_\ell(n)))$ where $F$ is a polynomial, $1\leq
q_1(n)<\cdots <q_\ell(n)$ are integer valued increasing functions satisfying
certain conditions and $X(n),\, n\geq 0$ is a sequence of independent random
variables with heavy tails.
