Tails of polynomials of random variables and stable limits for nonconventional sums

Abstract

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 ${\sum }_{Nt\ge n\ge 1}F(X(q_1(n)),...,X(q_{\ell }(n)))$ where $F$ is a polynomial, $1\le q_1(n)<\cdots <q_{\ell }(n)$ are integer valued increasing functions satisfying certain conditions and $X(n),n\ge 0$ is a sequence of independent random variables with heavy tails.

arXiv comment25 pages The paper is withdrawn since Theorem 2.3 holds true as stated only for a certain class of polynomials while, in general, only convergence of finite dimensional distributions can be proved. Soon a corrected version will be submitted