Limit distributions for polynomials with independent and identically distributed entries

TL;DR

This paper characterizes the limiting distributions of polynomial functions of i.i.d. variables with bounded degree, showing they can be represented as polynomials involving Gaussian variables, and introduces new tools for analyzing convergence.

Contribution

It provides a comprehensive characterization of limit laws for bounded-degree polynomials of i.i.d. variables, including Gaussian and non-Gaussian cases, with novel criteria for convergence.

Findings

Limiting distributions can be expressed as polynomials involving Gaussian variables.

Introduces a new criterion for central convergence based on maximal directional influence.

Develops techniques for asymptotic independence and dimensional reduction.

Abstract

We characterize the limiting distributions of random variables of the form $P_n\left( (X_i)_{i \ge 1} \right)$, where: (i) $(P_n)_{n \ge 1}$ is a sequence of multivariate polynomials, each potentially involving countably many variables; (ii) there exists a constant $D \ge 1$ such that for all $n \ge 1$, the degree of $P_n$ is bounded above by $D$; (iii) $(X_i)_{i \ge 1}$ is a sequence of independent and identically distributed random variables, each with zero mean, unit variance, and finite moments of all orders. More specifically, we prove that the limiting distributions of these random variables can always be represented as the law of $P_\infty\left( (X_i, G_i)_{i \ge 1} \right)$, where $P_\infty$ is a polynomial of degree at most $D$ (potentially involving countably many variables), and $(G_i)_{i \ge 1}$ is a sequence of independent standard Gaussian random variables, which is independent of $(X_i)_{i \ge 1}$. We solve this problem in full generality, addressing both Gaussian and non-Gaussian inputs, and with no extra assumption on the coefficients of the polynomials. In the Gaussian case, our proof builds upon several original tools of independent interest, including a new criterion for central convergence based on the concept of maximal directional influence. Beyond asymptotic normality, this novel notion also enables us to derive quantitative bounds on the degree of the polynomial representing the limiting law. We further develop techniques regarding asymptotic independence and dimensional reduction. To conclude for polynomials with non-Gaussian inputs, we combine our findings in the Gaussian case with invariance principles.