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

TL;DR

This paper establishes decay rates for tail probabilities of multivariate polynomials of heavy-tailed independent variables and proves stable limit theorems for nonconventional sums involving such polynomials.

Contribution

It introduces new decay rate results for tail probabilities and derives stable limit theorems for complex nonconventional sums of heavy-tailed variables.

Findings

Decay rates for tail probabilities of polynomial functions.

Stable limit theorems for nonconventional sums.

Applicability to heavy-tailed independent variables.

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 $\sum_{Nt\geq n\geq 1}F(X(q_1(n)),...,X(q_\ell(n)))$ 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.