Asymptotic Distribution of Bernoulli Quadratic Forms

TL;DR

This paper characterizes the limiting distribution of Bernoulli quadratic forms in sparse regimes, revealing a decomposition into Poisson-related components and establishing universality and conditions for Poisson convergence.

Contribution

It provides the first comprehensive decomposition theorem for the asymptotic distribution of Bernoulli quadratic forms in sparse settings, including universality and Poisson convergence criteria.

Findings

Limiting distribution decomposes into Poisson-related components.

Universality extends results to various discrete distributions.

Identifies necessary and sufficient conditions for Poisson convergence.

Abstract

Consider the random quadratic form $T_n=\sum_{1 \leq u < v \leq n} a_{uv} X_u X_v$, where $((a_{uv}))_{1 \leq u, v \leq n}$ is a $\{0, 1\}$-valued symmetric matrix with zeros on the diagonal, and $X_1,$ $X_2, \ldots, X_n$ are i.i.d. $\mathrm{Ber}(p_n)$. In this paper, we prove various characterization theorems about the limiting distribution of $T_n$, in the sparse regime, where $0 < p_n \ll 1$ such that $\mathbb E(T_n)=O(1).$ The main result is a decomposition theorem showing that distributional limits of $T_n$ is the sum of three components: a mixture which consists of a quadratic function of independent Poisson variables; a linear Poisson mixture, where the mean of the mixture is itself a (possibly infinite) linear combination of independent Poisson random variables; and another independent Poisson component. This is accompanied with a universality result which allows us to replace the Bernoulli distribution with a large class of other discrete distributions. Another consequence of the general theorem is a necessary and sufficient condition for Poisson convergence, where an interesting second moment phenomenon emerges.