Nonconventional Polynomial CLT

TL;DR

This paper establishes a functional central limit theorem for sums involving polynomially indexed functions of stochastic processes, extending classical CLT results to nonconventional polynomial settings.

Contribution

It introduces a new functional CLT for sums with polynomial indices, broadening the scope of limit theorems in stochastic process theory.

Findings

Proves a functional CLT for polynomial-indexed sums

Extends classical CLT to nonconventional polynomial cases

Provides a framework for analyzing polynomial-dependent stochastic sums

Abstract

We obtain a functional central limit theorem (CLT) for sums of the form $\xi_N(t)=\frac1{\sqrt N}\sum_{n=1}^{[Nt]}\big(F(X(q_1(n)),...,X(q_\ell(n)))-\bar F\big)$ where $q_1,...,q_\ell$ are polynomials.