The universality of homogeneous polynomial forms and critical limits

TL;DR

This paper extends a universality result for homogeneous polynomial forms in i.i.d. variables, showing the convergence to normality holds under broader conditions, and applies it to establish a CLT for dependent nonlinear processes at the boundary of short and long memory.

Contribution

It extends the universality of polynomial forms to infinite terms and applies this to prove a CLT for boundary-dependent nonlinear processes.

Findings

Universality extends to infinite polynomial forms.

Contraction criterion enables CLT for boundary memory processes.

Normal convergence holds under broader variable conditions.

Abstract

Nourdin et al. [9] established the following universality result: if a sequence of off-diagonal homogeneous polynomial forms in i.i.d. standard normal random variables converges in distribution to a normal, then the convergence also holds if one replaces these i.i.d. standard normal random variables in the polynomial forms by any independent standardized random variables with uniformly bounded third absolute moment. The result, which was stated for polynomial forms with a finite number of terms, can be extended to allow an infinite number of terms in the polynomial forms. Based on a contraction criterion derived from this extended universality result, we prove a central limit theorem for a strongly dependent nonlinear processes, whose memory parameter lies at the boundary between short and long memory.